arithmetic progression 2

Arithmetic progressionFrom Wikipedia, the free encyclopedia

Contents

1 Arithmetic progression 11.1 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Formulas at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Geometric progression 52.1 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Innite geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Relationship to geometry and Euclids work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Geometric series 133.1 Common ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.3 Proof of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.4 Generalized formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.1 Repeating decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

ii CONTENTS

3.3.2 Archimedes quadrature of the parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.3 Fractal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.4 Zenos paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.5 Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.6 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.7 Geometric power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.1 Specic geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5.1 History and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5.2 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5.3 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Scale factor 254.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Sequence 265.1 Examples and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.3 Specifying a sequence by recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Formal denition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2.2 Finite and innite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.3 Increasing and decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.4 Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.5 Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3.1 Denition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3.2 Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3.3 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5 Use in other elds of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.5.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.5.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.5.4 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.5.5 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5.6 Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5.7 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

CONTENTS iii

5.6 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.7 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.8 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.12 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.12.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.12.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.12.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 1

Arithmetic progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that thedierence between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 is an arithmeticprogression with common dierence of 2.If the initial term of an arithmetic progression is a1 and the common dierence of successive members is d, then thenth term of the sequence ( an ) is given by:

an = a1 + (n 1)d;and in general

an = am + (nm)d:A nite portion of an arithmetic progression is called a nite arithmetic progression and sometimes just called anarithmetic progression. The sum of a nite arithmetic progression is called an arithmetic series.The behavior of the arithmetic progression depends on the common dierence d. If the common dierence is:

Positive, the members (terms) will grow towards positive innity. Negative, the members (terms) will grow towards negative innity.

1.1 SumThis section is about Finite arithmetic series. For Innite arithmetic series, see Innite arithmetic series.

Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, theresulting sequence has a single repeated value in it, equal to the sum of the rst and last numbers (2 + 14 = 16). Thus16 5 = 80 is twice the sum.The sum of the members of a nite arithmetic progression is called an arithmetic series. For example, consider thesum:

2 + 5 + 8 + 11 + 14

This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of therst and last number in the progression (here 2 + 14 = 16), and dividing by 2:

n(a1 + an)

2

1

2 CHAPTER 1. ARITHMETIC PROGRESSION

In the case above, this gives the equation:

2 + 5 + 8 + 11 + 14 =5(2 + 14)

2=

5 162

= 40:

This formula works for any real numbers a1 and an . For example:

32

+

12

+

1

2=

332 + 12

2= 3

2:

1.1.1 DerivationTo derive the above formula, begin by expressing the arithmetic series in two dierent ways:

Sn = a1 + (a1 + d) + (a1 + 2d) + + (a1 + (n 2)d) + (a1 + (n 1)d)Sn = (an (n 1)d) + (an (n 2)d) + + (an 2d) + (an d) + an:Adding both sides of the two equations, all terms involving d cancel:

2Sn = n(a1 + an):

Dividing both sides by 2 produces a common form of the equation:

Sn =n

2(a1 + an):

An alternate form results from re-inserting the substitution: an = a1 + (n 1)d :

Sn =n

2[2a1 + (n 1)d]:

Furthermore the mean value of the series can be calculated via: Sn/n :

n =a1 + an

2:

In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics andIndian astronomy, gave this method in the Aryabhatiya (section 2.18).

1.2 ProductThe product of the members of a nite arithmetic progression with an initial element a1, common dierences d, andn elements in total is determined in a closed expression

a1a2 an = da1dd(a1d

+ 1)d(a1d

+ 2) d(a1d

+ n 1) = dna1d

n= dn

(a1/d+ n)

(a1/d);

where xn denotes the rising factorial and denotes the Gamma function. (Note however that the formula is not validwhen a1/d is a negative integer or zero.)This is a generalization from the fact that the product of the progression 1 2 n is given by the factorial n!and that the product

1.3. STANDARD DEVIATION 3

m (m+ 1) (m+ 2) (n 2) (n 1) nfor positive integersm and n is given by

n!

(m 1)! :

Taking the example from above, the product of the terms of the arithmetic progression given by an = 3 + (n1)(5)up to the 50th term is

P50 = 550 (3/5 + 50)

(3/5) 3:78438 1098:

1.3 Standard deviationThe standard deviation of any arithmetic progression can be calculated via:

= jdjr

(n 1)(n+ 1)12

where n is the number of terms in the progression, and d is the common dierence between terms

1.4 IntersectionsThe intersection of any two doubly-innite arithmetic progressions is either empty or another arithmetic progression,which can be found using the Chinese remainder theorem. If each two progressions in a family of doubly-innitearithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is,innite arithmetic progressions form a Helly family.[1] However, the intersection of innitely many innite arithmeticprogressions might be a single number rather than itself being an innite progression.

1.5 Formulas at a GlanceIf

a1

an

d

n

Sn

n

then

an = a1 + (n 1)d;an = am + (nm)d:Sn =

n

2[2a1 + (n 1)d]:

Sn =n(a1 + an)

2

4 CHAPTER 1. ARITHMETIC PROGRESSION

5. n = Sn/n

n =a1 + an

2:

1.6 See also Arithmetico-geometric sequence Generalized arithmetic progression - is a set of integers constructed as an arithmetic progression is, but allowingseveral possible dierences.

Harmonic progression Heronian triangles with sides in arithmetic progression Problems involving arithmetic progressions Utonality

1.7 References[1] Duchet, Pierre (1995), Hypergraphs, in Graham, R. L.; Grtschel, M.; Lovsz, L., Handbook of combinatorics, Vol. 1,

2, Amsterdam: Elsevier, pp. 381432, MR 1373663. See in particular Section 2.5, Helly Property, pp. 393394.

Sigler, Laurence E. (trans.) (2002). Fibonaccis Liber Abaci. Springer-Verlag. pp. 259260. ISBN 0-387-95419-8.

1.8 External links Hazewinkel, Michiel, ed. (2001), Arithmetic series, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Weisstein, Eric W., Arithmetic progression, MathWorld. Weisstein, Eric W., Arithmetic series, MathWorld.

Chapter 2

Geometric progression

Diagram illustrating three basic geometric sequences of the pattern 1(rn1) up to 6 iterations deep. The rst block is a unit block andthe dashed line represents the innite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3respectively.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers whereeach term after the rst is found by multiplying the previous one by a xed, non-zero number called the commonratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5,2.5, 1.25, ... is a geometric sequence with common ratio 1/2.Examples of a geometric sequence are powers rk of a xed number r, such as 2k and 3k. The general form of ageometric sequence is

a; ar; ar2; ar3; ar4; : : :

where r 0 is the common ratio and a is a scale factor, equal to the sequences start value.

5

6 CHAPTER 2. GEOMETRIC PROGRESSION

2.1 Elementary propertiesThe n-th term of a geometric sequence with initial value a and common ratio r is given by

an = a rn1:

Such a geometric sequence also follows the recursive relation

an = r an1 for every integer n 1:

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in thesequence all have the same ratio.The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbersswitching from positive to negative and back. For instance

1, 3, 9, 27, 81, 243, ...

is a geometric sequence with common ratio 3.The behaviour of a geometric sequence depends on the value of the common ratio.If the common ratio is:

Positive, the terms will all be the same sign as the initial term.

Negative, the terms will alternate between positive and negative.

Greater than 1, there will be exponential growth towards positive or negative innity (depending on the sign ofthe initial term).

1, the progression is a constant sequence.

Between 1 and 1 but not zero, there will be exponential decay towards zero.

1, the progression is an alternating sequence

Less than 1, for the absolute values there is exponential growth towards (unsigned) innity, due to the alter-nating sign.

Geometric sequences (with common ratio not equal to 1, 1 or 0) show exponential growth or exponential decay, asopposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, (with commondierence 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population.Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields ageometric progression, while taking the logarithm of each term in a geometric progression with a positive commonratio yields an arithmetic progression.An interesting result of the denition of a geometric progression is that for any value of the common ratio, any threeconsecutive terms a, b and c will satisfy the following equation:

b2 = ac

where b is considered to be the geometric mean between a and c.

2.2. GEOMETRIC SERIES 7

2.2 Geometric seriesComputation of the sum 2 + 10 + 50 + 250. The sequence is multiplied term by term by 5, and then subtracted fromthe original sequence. Two terms remain: the rst term, a, and the term one beyond the last, or arm. The desiredresult, 312, is found by subtracting these two terms and dividing by 1 5.A geometric series is the sum of the numbers in a geometric progression. For example:

2 + 10 + 50 + 250 = 2 + 2 5 + 2 52 + 2 53:Letting a be the rst term (here 2), m be the number of terms (here 4), and r be the constant that each term ismultiplied by to get the next term (here 5), the sum is given by:

a(1 rm)1 r

In the example above, this gives:

2 + 10 + 50 + 250 =2(1 54)1 5 =

12484 = 312:

The formula works for any real numbers a and r (except r = 1, which results in a division by zero). For example:

2 + 42 83 = 2 + (2)2 + (2)3 = 2(1 (2)3)

1 (2) =2(1 + 83)

1 + 2 214:855:

2.2.1 DerivationTo derive this formula, rst write a general geometric series as:

nXk=1

ark1 = ar0 + ar1 + ar2 + ar3 + + arn1:

We can nd a simpler formula for this sum by multiplying both sides of the above equation by 1 r, and we'll seethat

(1 r)nX

k=1

ark1 = (1 r)(ar0 + ar1 + ar2 + ar3 + + arn1)

= ar0 + ar1 + ar2 + ar3 + + arn1 ar1 ar2 ar3 arn1 arn= a arn

since all the other terms cancel. If r 1, we can rearrange the above to get the convenient formula for a geometricseries that computes the sum of n terms:

nXk=1

ark1 =a(1 rn)1 r :

2.2.2 Related formulasIf one were to begin the sum not from k=0, but from a dierent value, say m, then


nXk=m

ark =a(rm rn+1)

1 r :

Dierentiating this formula with respect to r allows us to arrive at formulae for sums of the form

nXk=0

ksrk:

For example:

d

dr

nXk=0

rk =nX

k=1

krk1 =1 rn+1(1 r)2

(n+ 1)rn

1 r :

For a geometric series containing only even powers of r multiply by 1 r2 :

(1 r2)nX

k=0

ar2k = a ar2n+2:

Then

nXk=0

ar2k =a(1 r2n+2)

1 r2 :

Equivalently, take r2 as the common ratio and use the standard formulation.For a series with only odd powers of r

(1 r2)nX

k=0

ar2k+1 = ar ar2n+3

and

nXk=0

ar2k+1 =ar(1 r2n+2)

1 r2 :

2.2.3 Innite geometric seriesMain article: Geometric series

An innite geometric series is an innite series whose successive terms have a common ratio. Such a series convergesif and only if the absolute value of the common ratio is less than one (|r| < 1). Its value can then be computed fromthe nite sum formulae

1Xk=0

ark = limn!1

nXk=0

ark = limn!1

a(1 rn+1)1 r =

a

1 r limn!1arn+1

1 r

Since:

2.2. GEOMETRIC SERIES 9

1

1/2

1/41/8

1/161/32

1/641/128

Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + which converges to 2.

rn+1 ! 0 as n!1 when jrj < 1:Then:

1Xk=0

ark =a

1 r 0 =a

1 r

For a series containing only even powers of r ,

1Xk=0

ar2k =a

1 r2

and for odd powers only,

1Xk=0

ar2k+1 =ar

1 r2

In cases where the sum does not start at k = 0,

1Xk=m

ark =arm

1 r

The formulae given above are valid only for |r| < 1. The latter formula is valid in every Banach algebra, as long as thenorm of r is less than one, and also in the eld of p-adic numbers if |r|p < 1. As in the case for a nite sum, we candierentiate to calculate formulae for related sums. For example,

d

dr

1Xk=0

rk =1Xk=0

krk1 =1

(1 r)2

This formula only works for |r| < 1 as well. From this, it follows that, for |r| < 1,


1Xk=0

krk =r

(1 r)2 ;1Xk=0

k2rk =r (1 + r)

(1 r)3 ;1Xk=0

k3rk =r1 + 4r + r2

(1 r)4

Also, the innite series 1/2 + 1/4 + 1/8 + 1/16 + is an elementary example of a series that converges absolutely.It is a geometric series whose rst term is 1/2 and whose common ratio is 1/2, so its sum is

1

2+

1

4+

1

8+

1

16+ = 1/2

1 (+1/2) = 1:

The inverse of the above series is 1/2 1/4 + 1/8 1/16 + is a simple example of an alternating series thatconverges absolutely.It is a geometric series whose rst term is 1/2 and whose common ratio is 1/2, so its sum is

1

2 1

4+

1

8 1

16+ = 1/2

1 (1/2) =1

3:

2.2.4 Complex numbers

The summation formula for geometric series remains valid even when the common ratio is a complex number. Inthis case the condition that the absolute value of r be less than 1 becomes that the modulus of r be less than 1. It ispossible to calculate the sums of some non-obvious geometric series. For example, consider the proposition

1Xk=0

sin(kx)rk

=r sin(x)

1 + r2 2r cos(x)

The proof of this comes from the fact that

sin(kx) = eikx eikx

2i;

which is a consequence of Eulers formula. Substituting this into the original series gives

1Xk=0

sin(kx)rk

=1

2i

" 1Xk=0

eix

r

k

1Xk=0

eix

r

k#

This is the dierence of two geometric series, and so it is a straightforward application of the formula for innitegeometric series that completes the proof.

2.3 ProductThe product of a geometric progression is the product of all terms. If all terms are positive, then it can be quicklycomputed by taking the geometric mean of the progressions rst and last term, and raising that mean to the powergiven by the number of terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence:take the arithmetic mean of the rst and last term and multiply with the number of terms.)

Qni=0 ar

i =p

a0 ann+1 (if a; r > 0 ).

2.4. RELATIONSHIP TO GEOMETRY AND EUCLIDS WORK 11

Proof:Let the product be represented by P:

P = a ar ar2 arn1 arn

Now, carrying out the multiplications, we conclude that

P = an+1r1+2+3++(n1)+(n)

Applying the sum of arithmetic series, the expression will yield

P = an+1rn(n+1)

2

P = (arn2 )n+1

We raise both sides to the second power:

P 2 = (a2rn)n+1 = (a arn)n+1

Consequently

P 2 = (a0 an)n+1

P = (a0 an)n+12

which concludes the proof.

2.4 Relationship to geometry and Euclids workBooks VIII and IX of Euclid's Elements analyzes geometric progressions (such as the powers of two, see the articlefor details) and give several of their properties.[1]

2.5 See also Arithmetic progression Arithmetico-geometric sequence Exponential function Harmonic progression Harmonic series Innite series Preferred number Thomas Robert Malthus Geometric distribution


2.6 References[1] Heath, Thomas L. (1956). The Thirteen Books of Euclids Elements (2nd ed. [Facsimile. Original publication:

Cambridge University Press, 1925] ed.). New York: Dover Publications.

Hall & Knight, Higher Algebra, p. 39, ISBN 81-8116-000-2

2.7 External links Hazewinkel, Michiel, ed. (2001), Geometric progression, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

Derivation of formulas for sum of nite and innite geometric progression at Mathalino.com Geometric Progression Calculator Nice Proof of a Geometric Progression Sum at sputsoft.com Weisstein, Eric W., Geometric Series, MathWorld.

Chapter 3

Geometric series

This article is about innite geometric series. For nite sums, see geometric progression.In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the

1/2

1/2

1/4

1/4

1/81/8

Each of the purple squares has 1/4 of the area of the next larger square (1/21/2 = 1/4, 1/41/4 = 1/16, etc.). The sum of theareas of the purple squares is one third of the area of the large square.

series

13

14 CHAPTER 3. GEOMETRIC SERIES

1

2+

1

4+

1

8+

1

16+

is geometric, because each successive term can be obtained by multiplying the previous term by 1/2.Geometric series are one of the simplest examples of innite series with nite sums, although not all of them havethis property. Historically, geometric series played an important role in the early development of calculus, and theycontinue to be central in the study of convergence of series. Geometric series are used throughout mathematics, andthey have important applications in physics, engineering, biology, economics, computer science, queueing theory,and nance.

3.1 Common ratio

The convergence of the geometric series with r=1/2 and a=1/2

1

1/2

1/4

1/8

1/161/32

The convergence of the geometric series with r=1/2 and a=1

The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the seriesis constant. This relationship allows for the representation of a geometric series using only two terms, r and a. Theterm r is the common ratio, and a is the rst term of the series. As an example the geometric series given in theintroduction,

12 +

14 +

18 +

116 +

3.2. SUM 15

may simply be written as

a+ ar + ar2 + ar3 + , with r = 12 and a = 12 .

The following table shows several geometric series with dierent common ratios:The behavior of the terms depends on the common ratio r:

If r is between 1 and +1, the terms of the series become smaller and smaller, approaching zero in thelimit and the series converges to a sum. In the case above, where r is one half, the series has the sumone.If r is greater than one or less than minus one the terms of the series become larger and larger inmagnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The seriesdiverges.)If r is equal to one, all of the terms of the series are the same. The series diverges.If r is minus one the terms take two values alternately (e.g. 2, 2, 2, 2, 2,... ). The sum of the termsoscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a dierent type of divergence and again theseries has no sum. See for example Grandis series: 1 1 + 1 1 + .

3.2 SumThe sum of a geometric series is nite as long as the absolute value of the ratio is less than 1; as the numbers nearzero, they become insignicantly small, allowing a sum to be calculated despite the series containing innitely-manyterms. The sum can be computed using the self-similarity of the series.

3.2.1 Example

A self-similar illustration of the sum s. Removing the largest circle results in a similar gure of 2/3 the original size.

Consider the sum of the following geometric series:

s = 1 +2

3+

4

9+

8

27+

This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the2/3 becomes a 4/9, and so on:

2

3s =

2

3+

4

9+

8

27+

16

81+


This new series is the same as the original, except that the rst term is missing. Subtracting the new series (2/3)s fromthe original series s cancels every term in the original but the rst:

s 23s = 1; so s = 3:

A similar technique can be used to evaluate any self-similar expression.

3.2.2 FormulaFor r 6= 1 , the sum of the rst n terms of a geometric series is:

a+ ar + ar2 + ar3 + + arn1 =n1Xk=0

ark = a1 rn1 r ;

where a is the rst term of the series, and r is the common ratio. We can derive this formula as follows:

Lets = a+ ar + ar2 + ar3 + + arn1:Thenrs = ar + ar2 + ar3 + ar4 + + arn

Thens rs = a arn

Thens(1 r) = a(1 rn); so s = a1 rn

1 r (ifr 6= 1):

As n goes to innity, the absolute value of r must be less than one for the series to converge. The sum then becomes

a+ ar + ar2 + ar3 + ar4 + =1Xk=0

ark =a

1 r ; for jrj < 1:

When a = 1, this can be simplied to:

1 + r + r2 + r3 + = 11 r ;

the left-hand side being a geometric series with common ratio r. We can derive this formula:

Lets = 1 + r + r2 + r3 + :Thenrs = r + r2 + r3 + :

Thens rs = 1; so s(1 r) = 1; thus and s = 11 r :

The general formula follows if we multiply through by a.The formula holds true for complex r, with the same restrictions (modulus of r is strictly less than one).

3.2.3 Proof of convergenceWe can prove that the geometric series converges using the sum formula for a geometric progression:

1 + r + r2 + r3 + = limn!1

1 + r + r2 + + rn

= limn!1

1 rn+11 r

3.2. SUM 17

Since (1 + r + r2 + ... + rn)(1r) = 1rn+1 and rn+1 0 for | r | < 1.Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series.Consider the function:

g(K) =rK

1 rNote that:

1 = g(0) g(1); r = g(1) g(2); r2 = g(2) g(3);

Thus:

S = 1 + r + r2 + r3 + ::: = (g(0) g(1)) + (g(1) g(2)) + (g(2) g(3)) +

If

jrj < 1

then

g(K) ! 0 asK !1

So S converges to

g(0) =1

1 r :

3.2.4 Generalized formulaFor r 6= 1 , the sum of the rst n terms of a geometric series is:

bXk=a

rk =ra rb+11 r ;

where a; b 2 N .We can derive this formula as follows:we put b = n 1) n = b+ 1

bXk=a

rk =n1Xk=0

rk a1Xk=0

rk

=1 rn1 r

1 ra1 r

=1 rn 1 + ra

1 r=

ra rb+11 r


3.3 Applications

3.3.1 Repeating decimalsMain article: Repeating decimal

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:

0:7777 : : : =7

10+

7

100+

7

1000+

7

10000+ :

The formula for the sum of a geometric series can be used to convert the decimal to a fraction:

0:7777 : : : =a

1 r =7/10

1 1/10 =7

9:

The formula works not only for a single repeating gure, but also for a repeating group of gures. For example:

0:123412341234 : : : =a

1 r =1234/10000

1 1/10000 =1234

9999:

Note that every series of repeating consecutive decimals can be conveniently simplied with the following:

0:09090909 : : : =09

99=

1

11:

0:143814381438 : : : =1438

9999:

0:9999 : : : =9

9= 1:

That is, a repeating decimal with repeat length n is equal to the quotient of the repeating part (as an integer) and 10n- 1.

3.3.2 Archimedes quadrature of the parabolaMain article: The Quadrature of the Parabola

Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. Hismethod was to dissect the area into an innite number of triangles.Archimedes Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 thearea of a green triangle, and so forth.Assuming that the blue triangle has area 1, the total area is an innite sum:

1 + 2

1

8

+ 4

1

8

2+ 8

1

8

3+ :

The rst term represents the area of the blue triangle, the second term the areas of the two green triangles, the thirdterm the areas of the four yellow triangles, and so on. Simplifying the fractions gives

1 +1

4+

1

16+

1

64+ :

3.3. APPLICATIONS 19

Archimedes dissection of a parabolic segment into innitely many triangles

This is a geometric series with common ratio 1/4 and the fractional part is equal to

1Xn=0

4n = 1 + 41 + 42 + 43 + = 43:

The sum is

1

1 r =1

1 14=

4

3:

This computation uses the method of exhaustion, an early version of integration. In modern calculus, the same areacould be found using a denite integral.

3.3.3 Fractal geometry

In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar gure.For example, the area inside the Koch snowake can be described as the union of innitely many equilateral triangles(see gure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and thereforehas exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking theblue triangle as a unit of area, the total area of the snowake is

1 + 3

1

9

+ 12

1

9

2+ 48

1

9

3+ :

The rst term of this series represents the area of the blue triangle, the second term the total area of the three greentriangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this seriesis geometric with constant ratio r = 4/9. The rst term of the geometric series is a = 3(1/9) = 1/3, so the sum is


The interior of the Koch snowake is a union of innitely many triangles.

1 +a

1 r = 1 +13

1 49=

8

5:

Thus the Koch snowake has 8/5 of the area of the base triangle.

3.3.4 Zenos paradoxesMain article: Zenos paradoxes

The convergence of a geometric series reveals that a sum involving an innite number of summands can indeed benite, and so allows one to resolve many of Zeno's paradoxes. For example, Zenos dichotomy paradox maintains

3.3. APPLICATIONS 21

that movement is impossible, as one can divide any nite path into an innite number of steps wherein each step istaken to be half the remaining distance. Zenos mistake is in the assumption that the sum of an innite number ofnite steps cannot be nite. This is of course not true, as evidenced by the convergence of the geometric series withr = 1/2 .

3.3.5 EuclidBook IX, Proposition 35[1] of Euclids Elements expresses the partial sum of a geometric series in terms of membersof the series. It is equivalent to the modern formula.

3.3.6 EconomicsMain article: Time value of money

In economics, geometric series are used to represent the present value of an annuity (a sum of money to be paid inregular intervals).For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end ofthe year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannotinvest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + I), where I is the yearly interest rate.Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + I )2 (squared because twoyears worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100per year in perpetuity is

1Xn=1

$100

(1 + I)n;

which is the innite series:

$100

(1 + I)+

$100

(1 + I)2+

$100

(1 + I)3+

$100

(1 + I)4+ :

This is a geometric series with common ratio 1 / (1 + I ). The sum is the rst term divided by (one minus the commonratio):

$100/(1 + I)

1 1/(1 + I) =$100

I:

For example, if the yearly interest rate is 10% ( I = 0.10), then the entire annuity has a present value of $100 / 0.10= $1000.This sort of calculation is used to compute the APR of a loan (such as a mortgage loan). It can also be used to estimatethe present value of expected stock dividends, or the terminal value of a security.

3.3.7 Geometric power seriesThe formula for a geometric series

1

1 x = 1 + x+ x2 + x3 + x4 +

can be interpreted as a power series in the Taylors theorem sense, converging where jxj < 1 . From this, one canextrapolate to obtain other power series. For example,


tan1(x) =Z

dx

1 + x2

=

Zdx

1 (x2)=

Z 1 +

x2+ x22 + x23 + dx=

Z 1 x2 + x4 x6 + dx

= x x3

3+x5

5 x

7

7+

=1Xn=0

(1)n2n+ 1

x2n+1

By dierentiating the geometric series, one obtains the variant[2]

1Xn=1

nxn1 =1

(1 x)2 for jxj < 1:

Similarly obtained are:

1Xn=2

n(n 1)xn2 = 2(1 x)3 for jxj < 1;

1Xn=3

n(n 1)(n 2)xn3 = 6(1 x)4 for jxj < 1:

3.4 See also 0.999... Asymptote Divergent geometric series Generalized hypergeometric function Geometric progression Neumann series Ratio test Root test Series (mathematics) Tower of Hanoi

3.4.1 Specic geometric series Grandis series: 1 1 + 1 1 + 1 + 2 + 4 + 8 + 1 2 + 4 8 +

3.5. REFERENCES 23

1/2 + 1/4 + 1/8 + 1/16 +

1/2 1/4 + 1/8 1/16 +

1/4 + 1/16 + 1/64 + 1/256 +

3.5 References[1] Euclids Elements, Book IX, Proposition 35. Aleph0.clarku.edu. Retrieved 2013-08-01.

[2] Taylor, Angus E. (1955), Advanced Calculus, Blaisdell, p. 603

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 278-279, 1985.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.

Courant, R. and Robbins, H. The Geometric Progression. 1.2.3 in What Is Mathematics?: An ElementaryApproach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 13-14, 1996.

Pappas, T. Perimeter, Area & the Innite Series. The Joy of Mathematics. San Carlos, CA: Wide WorldPubl./Tetra, pp. 134-135, 1989.

James Stewart (2002). Calculus, 5th ed., Brooks Cole. ISBN 978-0-534-39339-7

Larson, Hostetler, and Edwards (2005). Calculus with Analytic Geometry, 8th ed., Houghton Miin Company.ISBN 978-0-618-50298-1

Roger B. Nelsen (1997). Proofs without Words: Exercises in Visual Thinking, The Mathematical Associationof America. ISBN 978-0-88385-700-7

Andrews, George E. (1998). The geometric series in calculus. The American Mathematical Monthly (Math-ematical Association of America) 105 (1): 3640. doi:10.2307/2589524. JSTOR 2589524.

3.5.1 History and philosophy

C. H. Edwards, Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer. ISBN 978-0-387-94313-8.

Swain, Gordon and Thomas Dence (April 1998). Archimedes Quadrature of the Parabola Revisited. Math-ematics Magazine 71 (2): 12330. doi:10.2307/2691014. JSTOR 2691014.

Eli Maor (1991). To Innity and Beyond: A Cultural History of the Innite, Princeton University Press. ISBN978-0-691-02511-7

Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN978-0-415-22526-7

3.5.2 Economics

Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company. ISBN978-0-393-95733-4

Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0-415-26784-7


3.5.3 Biology Edward Batschelet (1992). Introduction to Mathematics for Life Scientists, 3rd ed., Springer. ISBN 978-0-387-09648-3

Richard F. Burton (1998). Biology by Numbers: An Encouragement to Quantitative Thinking, Cambridge Uni-versity Press. ISBN 978-0-521-57698-7

3.5.4 Computer science John Rast Hubbard (2000). Schaums Outline of Theory and Problems of Data Structures With Java, McGraw-Hill. ISBN 978-0-07-137870-3

3.6 External links Hazewinkel, Michiel, ed. (2001), Geometric progression, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

Weisstein, Eric W., Geometric Series, MathWorld. Geometric Series at PlanetMath.org. Peppard, Kim. College Algebra Tutorial on Geometric Sequences and Series. West Texas A&MUniversity. Casselman, Bill. A Geometric Interpretation of the Geometric Series (Applet). Geometric Series by Michael Schreiber, Wolfram Demonstrations Project, 2007.

Chapter 4

Scale factor

A scale factor is a number which scales, or multiplies, some quantity. In the equation y = Cx, C is the scale factorfor x. C is also the coecient of x, and may be called the constant of proportionality of y to x. For example, doublingdistances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scalefactor of one half. The basic equation for it is image over preimage.In the eld of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio ofany two corresponding lengths in two similar geometric gures is also called a scale factor.

4.1 See also Scale (ratio) Scale (map) Scales of scale models Scaling (geometry) Scalar (mathematics) Scaling in gravity Scaling in statistical estimation Scale factor (computer science) Scale factor (cosmology) Orthogonal coordinates

25

Chapter 5

Sequence

Sequential redirects here. For the manual transmission, see Sequential manual transmission. For other uses, seeSequence (disambiguation).

In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it containsmembers (also called elements, or terms). The number of elements (possibly innite) is called the length of thesequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at dierent positionsin the sequence. Formally, a sequence can be dened as a function whose domain is a countable totally ordered set,such as the natural numbers.For example, (M, A, R, Y) is a sequence of letters with the letter 'M' rst and 'Y' last. This sequence diers from(A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two dierent positions, is avalid sequence. Sequences can be nite, as in these examples, or innite, such as the sequence of all even positiveintegers (2, 4, 6,...). In computing and computer science, nite sequences are sometimes called strings, words or lists,the dierent names commonly corresponding to dierent ways to represent them into computer memory; innitesequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may beexcluded depending on the context.

An innite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. Itis, however, bounded.

26

5.1. EXAMPLES AND NOTATION 27

5.1 Examples and notationA sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number ofmathematical disciplines for studying functions, spaces, and other mathematical structures using the convergenceproperties of sequences. In particular, sequences are the basis for series, which are important in dierential equationsand analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in thestudy of prime numbers.There are a number of ways to denote a sequence, some of which are more useful for specic types of sequences.One way to specify a sequence is to list the elements. For example, the rst four odd numbers form the sequence(1,3,5,7). This notation can be used for innite sequences as well. For instance, the innite sequence of positiveodd integers can be written (1,3,5,7,...). Listing is most useful for innite sequences with a pattern that can be easilydiscerned from the rst few elements. Other ways to denote a sequence are discussed after the examples.

5.1.1 Important examples

3 2

1 1

5

8

A tiling with squares whose sides are successive Fibonacci numbers in length.

There are many important integer sequences. The prime numbers are the natural numbers bigger than 1, that haveno divisors but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). Thestudy of prime numbers has important applications for mathematics and specically number theory.The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The rsttwo elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...).Other interesting sequences include the ban numbers, whose spellings do not contain a certain letter of the alphabet.For instance, the eban numbers (do not contain 'e') form the sequence (2,4,6,30,32,34,36,40,42,...). Another sequencebased on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...).For a list of important examples of integers sequences see On-line Encyclopedia of Integer Sequences.Other important examples of sequences include ones made up of rational numbers, real numbers, and complex num-bers. The sequence (.9,.99,.999,.9999,...) approaches the number 1. In fact, every real number can be written asthe limit of a sequence of rational numbers. It is this fact that allows us to write any real number as the limit of asequence of decimals. For instance, is the limit of the sequence (3,3.1,3.14,3.141,3.1415,...). The sequence for ,however, does not have any pattern that is easily discernible by eye, unlike the sequence (0.9,0.99,...).

28 CHAPTER 5. SEQUENCE

5.1.2 IndexingOther notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not havea pattern such as the digits of . This section focuses on the notations used for sequences that are a map from a subsetof the natural numbers. For generalizations to other countable index sets see the following section and below.The terms of a sequence are commonly denoted by a single variable, say an, where the index n indicates the nthelement of the sequence.

a1 $ element 1sta2 $ element 2nda3 $ element 3rd... ...

an1 $ element (n-1)than $ element nth

an+1 $ element (n+1)th... ...

Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whoseelements are related to the index n (the elements position) in a simple way. For instance, the sequence of the rst 10square numbers could be written as

(a1; a2; :::; a10); ak = k2:

This represents the sequence (1,4,9,...100). This notation is often simplied further as

(ak)10k=1; ak = k

2:

Here the subscript {k=1} and superscript 10 together tell us that the elements of this sequence are the ak such that k= 1, 2, ..., 10.Sequences can be indexed beginning and ending from any integer. The innity symbol is often used as the super-script to indicate the sequence including all integer k-values starting with a certain one. The sequence of all positivesquares is then denoted

(ak)1k=1; ak = k

2:

In cases where the set of indexing numbers is understood, such as in analysis, the subscripts and superscripts are oftenleft o. That is, one simply writes ak for an arbitrary sequence. In analysis, k would be understood to run from 1 to. However, sequences are often indexed starting from zero, as in

(ak)1k=0 = (a0; a1; a2; :::):

In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easilyinferred. In these cases the index set may be implied by a listing of the rst few abstract elements. For instance, thesequence of squares of odd numbers could be denoted in any of the following ways.

(1; 9; 25; :::) (a1; a3; a5; :::); ak = k2

(a2k1)1k=1; ak = k2

5.2. FORMAL DEFINITION AND BASIC PROPERTIES 29

(ak)1k=1; ak = (2k 1)2

((2k 1)2)1k=1

Moreover, the subscripts and superscripts could have been left o in the third, fourth, and fth notations if theindexing set was understood to be the natural numbers.Finally, sequences can more generally be denoted by writing a set inclusion in the subscript, such as in

(ak)k2N

The set of values that the index can take on is called the index set. In general, the ordering of the elements ak isspecied by the order of the elements in the indexing set. When N is the index set, the element ak+1 comes after theelement ak since in N, the element (k+1) comes directly after the element k.

5.1.3 Specifying a sequence by recursionSequences whose elements are related to the previous elements in a straightforward way are often specied usingrecursion. This is in contrast to the specication of sequence elements in terms of their position.To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones beforeit. In addition, enough initial elements must be specied so that new elements of the sequence can be specied bythe rule. The principle of mathematical induction can be used to prove that a sequence is well-dened, which is tosay that that every element of the sequence is specied at least once and has a single, unambiguous value. Inductioncan also be used to prove properties about a sequence, especially for sequences whose most natural specication isby recursion.The Fibonacci sequence can be dened using a recursive rule along with two initial elements. The rule is that eachelement is the sum of the previous two elements, and the rst two elements are 0 and 1.

an = an1 + an2 , with a0 = 0 and a1 = 1 .

The rst ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence thatis dened recursively is Recamans sequence, considered at the beginning of this section. We can dene Recamanssequence by

a0 = 0 and an = an1n if the result is positive and not already in the list. Otherwise, an = an1+n.

Not all sequences can be specied by a rule in the form of an equation, recursive or not, and some can be quitecomplicated. For example, the sequence of prime numbers is the set of prime numbers in their natural order. Thisgives the sequence (2,3,5,7,11,13,17,...).One can also notice that the next element of a sequence is a function of the element before, and so we can write thenext element as : an+1 = f(an)This functional notation can prove useful when one wants to prove the global monotony of the sequence.

5.2 Formal denition and basic propertiesThere are many dierent notions of sequences in mathematics, some of which (e.g., exact sequence) are not coveredby the denitions and notations introduced below.

5.2.1 Formal denitionA sequence is usually dened as a function whose domain is a countable totally ordered set, although in many disci-plines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset


of the natural numbers to the real numbers.[1] In other words, a sequence is a map f(n) : N R. To recover ourearlier notation we might identify an = f(n) for all n or just write an : N R.In complex analysis, sequences are dened as maps from the natural numbers to the complex numbers (C).[2] Intopology, sequences are often dened as functions from a subset of the natural numbers to a topological space.[3]Sequences are an important concept for studying functions and, in topology, topological spaces. An important gener-alization of sequences, called a net, is to functions from a (possibly uncountable) directed set to a topological space.

5.2.2 Finite and innite

The length of a sequence is dened as the number of terms in the sequence.A sequence of a nite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has noelements.Normally, the term innite sequence refers to a sequence which is innite in one direction, and nite in the otherthesequence has a rst element, but no nal element, it is called a singly innite, or one-sided (innite) sequence,when disambiguation is necessary. In contrast, a sequence that is innite in both directionsi.e. that has neither arst nor a nal elementis called a bi-innite sequence, two-way innite sequence, or doubly innite sequence.A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( , 4, 2,0, 2, 4, 6, 8 ), is bi-innite. This sequence could be denoted (2n)1n=1 .One can interpret singly innite sequences as elements of the semigroup ring of the natural numbers R[N], and doublyinnite sequences as elements of the group ring of the integers R[Z]. This perspective is used in the Cauchy productof sequences.

5.2.3 Increasing and decreasing

A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For asequence (an)1n=1 this can be written as an an for all n N. If each consecutive term is strictly greater than(>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonicallydecreasing if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasingif each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotonesequence. This is a special case of the more general notion of a monotonic function.The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoidany possible confusion with strictly increasing and strictly decreasing, respectively.

5.2.4 Bounded

If the sequence of real numbers (an) is such that all the terms, after a certain one, are less than some real numberM,then the sequence is said to be bounded from above. In less words, this means an M for all n greater than N forsome pair M and N. Any such M is called an upper bound. Likewise, if, for some real m, an m for all n greaterthan some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence isboth bounded from above and bounded from below then the sequence is said to be bounded.

5.2.5 Other types of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elementswithout disturbing the relative positions of the remaining elements. For instance, the sequence of positive evenintegers (2,4,6,...) is a subsequence of the positive integers (1,2,3,...). The positions of some elements change whenother elements are deleted. However, the relative positions are preserved.Some other types of sequences that are easy to dene include:

An integer sequence is a sequence whose terms are integers. A polynomial sequence is a sequence whose terms are polynomials.

5.3. LIMITS AND CONVERGENCE 31

A positive integer sequence is sometimes calledmultiplicative if anm = an am for all pairs n,m such that n andm are coprime.[4] In other instances, sequences are often called multiplicative if an = na1 for all n. Moreover,the multiplicative Fibonacci sequence satises the recursion relation an = an an.

5.3 Limits and convergenceMain article: Limit of a sequenceOne of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit.

5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

n + 12n2

The plot of a convergent sequence (a) is shown in blue. Visually we can see that the sequence is converging to the limit zero as nincreases.

Continuing informally, a (singly innite) sequence has a limit if it approaches some value L, called the limit, as nbecomes very large. That is, for an abstract sequence (an) (with n running from 1 to innity understood) the valueof an approaches L as n, denoted

limn!1 an = L:

More precisely, the sequence converges if there exists a limit L such that the remaining a's are arbitrarily close to Lfor some n large enough.If a sequence converges to some limit, then it is convergent; otherwise it is divergent.If an gets arbitrarily large as n we write

limn!1 an =1:


In this case we say that the sequence (an) diverges, or that it converges to innity.If an becomes arbitrarily small negative numbers (large in magnitude) as n we write

limn!1 an = 1

and say that the sequence diverges or converges to minus innity.

5.3.1 Denition of convergence

For sequences that can be written as (an)1n=1 with an R we can write (an) with the indexing set understood as N.These sequences are most common in real analysis. The generalizations to other types of sequences are consideredin the following section and the main page Limit of a sequence.Let (an) be a sequence. In words, the sequence (an) is said to converge if there exists a number L such that no matterhow close we want the an to be to L (say -close where > 0), we can nd a natural number N such that all terms(aN+1, aN+2, ...) are further closer to L (within of L). [1] This is often written more compactly using symbols. Forinstance,

for all > 0, there exists a natural number N such that L < an < L+ for all n N.

In even more compact notation

8 > 0; 9N 2 N s.t. 8n N; jan Lj < :

The dierence in the denitions of convergence for (one-sided) sequences in complex analysis and metric spaces isthat the absolute value |an L| is interpreted as the distance in the complex plane (

pzz ), and the distance under

the appropriate metric, respectively.

5.3.2 Applications and important results

Important results for convergence and limits of (one-sided) sequences of real numbers include the following. Theseequalities are all true at least when both sides exist. For a discussion of when the existence of the limit on one sideimplies the existence of the other see a real analysis text such as can be found in the references.[1][5]

The limit of a sequence is unique.

limn!1(an bn) = limn!1 an limn!1 bn limn!1 can = c limn!1 an limn!1(anbn) = (limn!1 an)(limn!1 bn)

limn!1 anbn = limn!1 anlimn!1 bn provided limn!1 bn 6= 0

limn!1 apn = [limn!1 an]p

If an bn for all n greater than some N, then limn!1 an limn!1 bn .

(Squeeze Theorem) If an cn bn for all n >N, and limn!1 an = limn!1 bn = L , then limn!1 cn = L.

If a sequence is bounded and monotonic then it is convergent.

A sequence is convergent if and only if every subsequence is convergent.

5.4. SERIES 33

The plot of a Cauchy sequence (X), shown in blue, asX versus n. Visually, we see that the sequence appears to be converging to thelimit zero as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence convergesto some limit.

5.3.3 Cauchy sequencesMain article: Cauchy sequenceA Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion ofa Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. Oneparticularly important result in real analysis is Cauchy characterization of convergence for sequences:

In the real numbers, a sequence is convergent if and only if it is Cauchy.

In contrast, in the rational numbers, e.g. the sequence dened by x1 = 1 and xn = xn + 2/xn/2 is Cauchy, but has norational limit, cf. here.

5.4 SeriesMain article: Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence. That is, adding the rst N terms of a (one-sided)sequence forms the Nth term of another sequence, called a series. Thus the N series of the sequence (a) results inanother sequence (SN) given by:

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3

... ...SN = a1 + a2 + a3 +

... ...

We can also write the nth term of the series as


SN =NXn=1

an:

Then the concepts used to talk about sequences, such as convergence, carry over to series (the sequence of partialsums) and the properties can be characterized as properties of the underlying sequences (such as (an) in the lastexample). The limit, if it exists, of an innite series (the series created from an innite sequence) is written as

limN!1

SN =

1Xn=1

an:

5.5 Use in other elds of mathematics

5.5.1 TopologySequence play an important role in topology, especially in the study of metric spaces. For instance:

A metric space is compact exactly when it is sequentially compact. A function from ametric space to another metric space is continuous exactly when it takes convergent sequencesto convergent sequences.

A metric space is a connected space if, whenever the space is partitioned into two sets, one of the two setscontains a sequence converging to a point in the other set.

A topological space is separable exactly when there is a dense sequence of points.

Sequences can be generalized to nets or lters. These generalizations allow one to extend some of the above theoremsto spaces without metrics.

Product topology

A product space of a sequence of topological spaces is the cartesian product of the spaces equipped with a naturaltopology called the product topology.More formally, given a sequence of spaces fXig , dene X such that

X :=Yi2I

Xi;

is the set of sequences fxig where each xi is an element of Xi . Let the canonical projections be written as pi :X Xi. Then the product topology on X is dened to be the coarsest topology (i.e. the topology with the fewestopen sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonotopology.

5.5.2 AnalysisIn analysis, when talking about sequences, one will generally consider sequences of the form

(x1; x2; x3; : : : ) or (x0; x1; x2; : : : )

which is to say, innite sequences of elements indexed by natural numbers.

5.5. USE IN OTHER FIELDS OF MATHEMATICS 35

It may be convenient to have the sequence start with an index dierent from 1 or 0. For example, the sequence denedby xn = 1/log(n) would be dened only for n 2. When talking about such innite sequences, it is usually sucient(and does not change much for most considerations) to assume that the members of the sequence are dened at leastfor all indices large enough, that is, greater than some given N.The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This typecan be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are oftenfunction spaces. Even more generally, one can study sequences with elements in some topological space.

Sequence spaces

Main article: Sequence space

A sequence space is a vector space whose elements are innite sequences of real or complex numbers. Equivalently, itis a function space whose elements are functions from the natural numbers to the eldK of real or complex numbers.The set of all such functions is naturally identied with the set of all possible innite sequences with elements in K,and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalarmultiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped witha norm, or at least the structure of a topological vector space.The most important sequences spaces in analysis are the p spaces, consisting of the p-power summable sequences,with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers.Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectivelydenoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwiseconvergence, under which it becomes a special kind of Frchet space called FK-space.

5.5.3 Linear algebraSequences over a eld may also be viewed as vectors in a vector space. Specically, the set of F-valued sequences(where F is a eld) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

5.5.4 Abstract algebraAbstract algebra employs several types of sequences, including sequences of mathematical objects such as groups orrings.

Free monoid

Main article: Free monoid

If A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the nitesequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroupA+ is the subsemigroup of A* containing all elements except the empty sequence.

Exact sequences

Main article: Exact sequence

In the context of group theory, a sequence

G0f1! G1 f2! G2 f3! fn! Gn

of groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to thekernel of the next:


im(fk) = ker(fk+1)

Note that the sequence of groups and homomorphisms may be either nite or innite.A similar denition can be made for certain other algebraic structures. For example, one could have an exact sequenceof vector spaces and linear maps, or of modules and module homomorphisms.

Spectral sequences

Main article: Spectral sequence

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groupsby taking successive approximations. Spectral sequences are a generalization of exact sequences, and since theirintroduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

5.5.5 Set theoryAn ordinal-indexed sequence is a generalization of a sequence. If is a limit ordinal and X is a set, an -indexedsequence of elements of X is a function from to X. In this terminology an -indexed sequence is an ordinarysequence.

5.5.6 ComputingAutomata or nite statemachines can typically be thought of as directed graphs, with edges labeled using some specicalphabet, . Most familiar types of automata transition from state to state by reading input letters from , followingedges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word).The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministicautomaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some inputletter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence ofsingle states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally usedto mean the latter.

5.5.7 StreamsInnite sequences of digits (or characters) drawn from a nite alphabet are of particular interest in theoretical com-puter science. They are often referred to simply as sequences or streams, as opposed to nite strings. Innite binarysequences, for instance, are innite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0,1} of all innite, binary sequences is sometimes called the Cantor space.An innite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalizationmethod for proofs.[6]

5.6 Types 1-sequence Arithmetic progression Cauchy sequence Farey sequence Fibonacci sequence

5.7. RELATED CONCEPTS 37

Geometric progression

Look-and-say sequence

ThueMorse sequence

5.7 Related concepts List (computing)

Ordinal-indexed sequence

Recursion (computer science)

Tuple

Set theory

5.8 Operations Cauchy product

Limit of a sequence

5.9 See also Enumeration

Net (topology) (a generalization of sequences)

On-Line Encyclopedia of Integer Sequences

Permutation

Recurrence relation

Sequence space

Set (mathematics)

5.10 References[1] Gaughan, Edward. 1.1 Sequences and Convergence. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.

[2] Edward B. Sa & Arthur David Snider (2003). Chapter 2.1. Fundamentals of Complex Analysis. ISBN 01-390-7874-6.

[3] James R. Munkres. Chapters 1&2. Topology. ISBN 01-318-1629-2.

[4] Lando, Sergei K. 7.4 Multiplicative sequences. Lectures on generating functions. AMS. ISBN 0-8218-3481-9.

[5] Dawikins, Paul. Series and Sequences. Pauls Online Math Notes/Calc II (notes). Retrieved 18 December 2012.

[6] Oazer, Kemal. FORMAL LANGUAGES, AUTOMATAANDCOMPUTATION: DECIDABILITY (PDF). cmu.edu.Carnegie-Mellon University. Retrieved 24 April 2015.


5.11 External links The dictionary denition of sequence at Wiktionary Hazewinkel, Michiel, ed. (2001), Sequence, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

The On-Line Encyclopedia of Integer Sequences Journal of Integer Sequences (free) Sequence at PlanetMath.org.

5.12. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 39

5.12 Text and image sources, contributors, and licenses5.12.1 Text

Arithmetic progression Source: https://en.wikipedia.org/wiki/Arithmetic_progression?oldid=671955716 Contributors: AxelBoldt, Tar-quin,WilliamAvery, Patrick, Chas zzz brown,Michael Hardy, UserGoogol, Andres, CharlesMatthews, Dcoetzee, Kbk, Hyacinth, McKay,Fredrik, Altenmann, Nikitadanilov, Giftlite, Jackol, CryptoDerk, Tbjablin, Discospinster, Murtasa, Goochelaar, Bobo192, Robotje, CS, .:Ajvol:., Haham hanuka, Msh210, Rh~enwiki, Snowolf, Terrible tony, Oleg Alexandrov, Hyperfusion, Sanjaymjoshi, Shreevatsa,GregorB, Cornince, Tbone, Salvatore Ingala, Chobot, Bgwhite, Siddhant, YurikBot, Wavelength, Hairy Dude, Icedemon, KSmrq, Bill-on-the-Hill, Petter Strandmark, Nick, Googl, Zzuuzz, Closedmouth, RDBury, Selfworm, InverseHypercube, Melchoir, Ohnoitsjamie,PrimeHunter, Octahedron80, DHN-bot~enwiki, Rrburke, Sebo.PL, Attys, Catapult, Aleenf1, Ckatz, 16@r, Euphrates~enwiki, MTS-bot~enwiki, ST47, Goldencako, Marek69, John254, Majorly, JAnDbot, Ricardo sandoval, VoABot II, JNW,David Eppstein, Nguyn HuDung, Ulfalizer, DarwinPeacock, Daniel5Ko, Policron, Milogardner, Jamesontai, Steel1943, VolkovBot, Johan1298~enwiki, TXiKiBoT,Anonymous Dissident, Ravig sagi, SieBot, Rlendog, Flyer22, Masgatotkaca, Harry~enwiki, Xhackeranywhere, Denisarona, Loren.wilton,Tai Chi Tech, ClueBot, DR23, Mild Bill Hiccup, Niceguyedc, Sohail555, DumZiBoT, XLinkBot, Gonzonoir, Kal-El-Bot, Addbot,Macarse, Delaszk, SamatBot, Numbo3-bot, Tide rolls, Legobot, Luckas-bot, Yobot, TaBOT-zerem, Mmxx, Writer on wiki, ,AnomieBOT, Ciphers, Rubinbot, Jim1138, 9258fahskh917fas, Flewis, Materialscientist, Citation bot, Donanayath, Xqbot, ncel Acar,RibotBOT, SassoBot, LuisVillegas, Sophus Bie, A.amitkumar, Dougofborg, RTFVerterra, Robo37, Pinethicket, SpaceFlight89, Ran-domStringOfCharacters, Dude1818, FoxBot, DixonDBot, Duoduoduo, KurtSchwitters, Jowa fan, EmausBot, Maschen, Nikunj Pandya,Petrb, ClueBot NG, Wcherowi, Widr, Princetct.007, Krenair, Walrus068, Kingsbrook, Sparkie82, Brad7777, Williamdemeo, Arpitkjain,Justincheng12345-bot, Scientic Alan 2, Jochen Burghardt, Razibot, Vshender, Zorch713, Chacho39, AtoiyonTayib, Velvel2, Sahil Rally,Arvindsingh0707, Sububu, Sourabh Tayade and Anonymous: 220

Geometric progression Source: https://en.wikipedia.org/wiki/Geometric_progression?oldid=670950573 Contributors: AxelBoldt, Zun-dark, Tarquin, Patrick, Michael Hardy, Delirium, Conti, Charles Matthews, Dcoetzee, Hyacinth, Fredrik, Matt me, R3m0t, May-ooranathan, Henrygb, Aetheling, Tobias Bergemann, Giftlite, Knutux, LucasVB, MarkSweep, Rpchase, Jcw69, Allefant, Moxfyre, MikeRosoft, MuDavid, Paul August, Aranel, Touriste, Elementalish, Aisaac, Msh210, Rh~enwiki, Arthena, PAR, Mosesofmason, Justinle-bar, Olethros, Gerbrant, Graham87, Yurik, Sango123, Lmatt, BradBeattie, Chobot, Sbrools, Redde, Siddhant, YurikBot, Wavelength,Icedemon, JabberWok, Dantheox, DarthVader, Haihe, Plamka, EAderhold, Lt-wiki-bot, Arthur Rubin, Nemu, Mike1024, Pred, Hearth,Banus, Thorney?, Finell, SmackBot, RDBury, Incnis Mrsi, Melchoir, Nereus124, Ixtli, Janmarthedal, Bluebot, Octahedron80, DHN-bot~enwiki, Can't sleep, clown will eat me, Jratt, Mark Wolfe, Nakon, Stefano85, Vina-iwbot~enwiki, Netnubie, Jim.belk, Advance512,Mets501, Pjrm, JForget, CmdrObot, Ichiroo, FilipeS, Haifadude, ST47, Goldencako, Tawkerbot4, Joeyfox10, Awmorp, Vanished Userjdksfajlasd, Thijs!bot, Dugwiki, AntiVandalBot, , JAnDbot, Leuko, Divyesikka, Gaeddal, 01001, MSBOT, James-BWatson, Meissmart, JJ Harrison, David Eppstein, Quanticle, Andylatto, Chrisalvino, Policron, DavidCBryant, DorganBot, Gp4rts,VolkovBot, Johan1298~enwiki, Je G., LokiClock, Philip Trueman, Af648, TXiKiBoT, Vertciel, SieBot, Yulu, Anchor Link Bot,Timeastor, ClueBot, Justin W Smith, DR23, Mathwizkid, He7d3r, Fattyjwoods, NellieBly, Addbot, Laubpatr, Metagraph, Fieldday-sunday, Tide rolls, Jarble, Odder, Luckas-bot, Yobot, , AnomieBOT, The Parting Glass, 9258fahskh917fas, Xqbot, Resident Mario,Krishano, LuisVillegas, Joxemai, RTFVerterra, Pepper, Gas Panic42, Amgc56, Turian, FoxBot, TobeBot, Thelema418, Nascar1996,Skittlestastegood, Jowa fan, EmausBot, Felix Homann, Wikipelli, Tahdah, Slawekb, Josve05a, L Kensington, Maschen, ClueBot NG,Wcherowi, PoqVaUSA, Ghostsarememories, Sparkie82, Rahaven, Brad7777, Anbu121, Arpitkjain, Henri.vanliempt, Amirki, 1Minow,Brirush, Dennis at Empa Media, Jhncls, Zereth, *thing goes, Hazo11413 and Anonymous: 221

Geometric series Source: https://en.wikipedia.org/wiki/Geometric_series?oldid=672472757 Contributors: AxelBoldt, Bryan Derksen,The Anome, XJaM, Heron, Michael Hardy, Willsmith, Pnm, ArnoLagrange, LittleDan, Poor Yorick, Jitse Niesen, Hyacinth, Henrygb,Per Abrahamsen, Vacuum, Giftlite, MSGJ, Vsb, Moxfyre, Rich Farmbrough, Guanabot, Paul August, Touriste, Kenyon, Mindma-trix, Salix alba, The wub, Nihiltres, Kri, Wavelength, Gillis, Closedmouth, Arthur Rubin, Reyk, Netrapt, Ghazer~enwiki, SmackBot,Michaelliv, Incnis Mrsi, InverseHypercube, Golwengaud, Kostmo, Hgrosser, Jbergquist, Black Carrot, Jim.belk, Happy-melon, Lavat-eraguy, CBM, Schaber, Arrataz, Gogo Dodo, Escarbot, Uplink3r, Thenub314, JamesBWatson, JJ Harrison, David Eppstein, JaGa,Ankitdoshi1, Eastmbr, R'n'B, Pbroks13, AstroHurricane001, Policron, Fylwind, Austinmohr, Pleasantville, LokiClock, Philip Trueman,Ocolon, Chenzw, StevenJohnston, Oboeboy, Caltas, Yerpo, T5j6p9, Archaeogenetics, Khvalamde, Shane87, Asperal, PerryTachett, Dr-garden, DonAByrd, ClueBot, Justin W Smith, DanielDeibler, Timberframe, Niceguyedc, Hans Adler, BOTarate, Eranus~enwiki, PCHS-NJROTC, HiTechHiTouch, Addbot, DOI bot, Zarcadia, Jarble, Clay Juicer, Yobot, AnomieBOT, Bdmy, Dithridge, Trut-h-urts man,Raamaiden, NOrbeck, Hugetim, Efadae, MrHeberRomo, Citation bot 1, S iliad, Hexadecachoron, Duoduoduo, Thelema418, Bobby122,WillNess, Ramblagir, Slawekb, 4blossoms, Souless194, VoilY'all, DASHBotAV, Mastomer, Rocketrod1960, ClueBot NG, Wcherowi,Helpful Pixie Bot, Jakemymath, Rahaven, Brad7777, , Zetazeros, OceanEngineerRI, Amirki, Webclient101, Saehry,Stephan Kulla, Frosty, Doctordubin, Hillbillyholiday, CsDix, Gkvp, Babitaarora, ColeLoki, Bellezzasolo, Staymathy, Vrkssai, Monkbot,Ktlabe, Tymon.r, Feitreim and Anonymous: 166

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Sequence Source: https://en.wikipedia.org/wiki/Sequence?oldid=673701777 Contributors: AxelBoldt, Mav, Zundark, Tarquin, XJaM,Toby Bartels, Imran, Camembert, Youandme, Lir, Patrick, Michael Hardy, Ihcoyc, Poor Yorick, Nikai, EdH, Charles Matthews, Dys-prosia, Greenrd, Hyacinth, Zero0000, Sabbut, Garo, Robbot, Lowellian, MathMartin, Stewartadcock, Henrygb, Bkell, Tosha, Centrx,Giftlite, BenFrantzDale, Lupin, Herbee, Horatio, Edcolins, Vadmium, Leonard Vertighel, Manuel Anastcio, Alexf, Fudo, Melikamp,


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Arithmetic progressionSumDerivation

ProductStandard deviationIntersectionsFormulas at a GlanceSee alsoReferences External links

Geometric progressionElementary propertiesGeometric seriesDerivationRelated formulasInfinite geometric seriesComplex numbers

ProductRelationship to geometry and Euclids workSee alsoReferencesExternal links

Geometric seriesCommon ratioSumExampleFormulaProof of convergenceGeneralized formula

ApplicationsRepeating decimalsArchimedes quadrature of the parabolaFractal geometryZenos paradoxesEuclidEconomicsGeometric power series

See alsoSpecific geometric series

ReferencesHistory and philosophyEconomicsBiologyComputer science

External links

Scale factorSee also

SequenceExamples and notation Important examplesIndexingSpecifying a sequence by recursion

Formal definition and basic properties Formal definitionFinite and infinite Increasing and decreasingBoundedOther types of sequences

Limits and convergenceDefinition of convergenceApplications and important resultsCauchy sequences

SeriesUse in other fields of mathematicsTopology AnalysisLinear algebra Abstract algebraSet theoryComputing Streams

TypesRelated conceptsOperationsSee alsoReferencesExternal linksText and image sources, contributors, and licensesTextImagesContent license

arithmetic progression 2

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