mate sequence

8/3/2019 MATE Sequence

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Sequence

In mathematics, a sequence is an ordered list of objects (or events). Like a set, it containsmembers (also called elements or terms), and the number of terms (possibly infinite) iscalled the length of the sequence. Unlike a set, order matters, and exactly the same

elements can appear multiple times at different positions in the sequence. A sequence is adiscrete function.

For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the orderingmatters. Sequences can be finite, as in this example, or infinite, such as the sequence ofall even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings orwords and infinite sequences as streams. The empty sequence ( ) is included in mostnotions of sequence, but may be excluded depending on the context.

Examples and notation

There are various and quite different notions of sequences in mathematics, some of which(e.g., exact sequence) are not covered by the notations introduced below.

In addition to identifying the elements of a sequence by their position, such as "the 3rdelement", elements may be given names for convenient referencing. For example asequence might be written as (a1, a2, a2, ), or (b0, b1, b2, ), or (c0, c2, c4, ),depending on what is useful in the application.

Finite and infinite

A more formal definition of a finite sequence with terms in a set S is a function from {1, 2,

..., n} to S for some n > 0. An infinite sequence in S is a function from {1, 2, ... } to S. Forexample, the sequence of prime numbers (2,3,5,7,11, ) is the function 12, 23, 35,47, 511, .

A sequence of a finite length n is also called an n-tuple. Finite sequences include theempty sequence ( ) that has no elements.

A function from all integers into a set is sometimes called a bi-infinite sequence or two-way infinite sequence. An example is the bi-infinite sequence of all even integers ( , -4,-2, 0, 2, 4, 6, 8 ).

Multiplicative

Let A = (a sequence defined by a function f:{1, 2, 3, ...} {1, 2, 3, ...}, such that a i =f(i).

The sequence is multiplicative if f(xy) = f(x)f(y) for all x,y such that x and y are coprime.[1]

Types and properties of sequences

A subsequence of a given sequence is a sequence formed from the given sequence bydeleting some of the elements without disturbing the relative positions of the remainingelements.
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If the terms of the sequence are a subset of an ordered set, then a monotonicallyincreasing sequence is one for which each term is greater than or equal to the term beforeit; if each term is strictly greater than the one preceding it, the sequence is called strictlymonotonically increasing. A monotonically decreasing sequence is defined similarly. Anysequence fulfilling the monotonicity property is called monotonic or monotone. This is aspecial case of the more general notion ofmonotonic function.

The terms nondecreasing and nonincreasing are used in order to avoid any possibleconfusion with strictly increasing and strictly decreasing, respectively.

If the terms of a sequence are integers, then the sequence is an integer sequence. If theterms of a sequence are polynomials, then the sequence is a polynomial sequence.

If S is endowed with a topology, then it becomes possible to consider convergence of aninfinite sequence in S. Such considerations involve the concept of the limit of a sequence.

If A is a set, the free monoid over A (denoted A*) is a monoid containing all the finite

sequences (or strings) of zero or more elements drawn from A, with the binary operation ofconcatenation. The free semigroup A+ is the subsemigroup of A* containing all elementsexcept the empty sequence.

Sequences in analysis

In analysis, when talking about sequences, one will generally consider sequences of theform

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

It may be convenient to have the sequence start with an index different from 1 or 0. Forexample, the sequence defined by xn = 1/log(n) would be defined only for n 2. Whentalking about such infinite sequences, it is usually sufficient (and does not change much formost considerations) to assume that the members of the sequence are defined at least forall indices large enough, that is, greater than some given N.

The most elementary type of sequences are numerical ones, that is, sequences of real orcomplex numbers. This type can be generalized to sequences of elements of some vectorspace. In analysis, the vector spaces considered are often function spaces. Even more

generally, one can study sequences with elements in some topological space.
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Arithmetic progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence ofnumbers such that the difference of any two successive members of the sequence is aconstant. For instance, the sequence 3, 5, 7, 9, 11, 13, is an arithmetic progression with

common difference 2.

If the initial term of an arithmetic progression is a1 and the common difference ofsuccessive members is d, then the nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression andsometimes just called an arithmetic progression.

The behavior of the arithmetic progression depends on the common difference d. If thecommon difference is:

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

SumThe sum of the members of a finite arithmetic progression is called an arithmetic series.

Expressing the arithmetic series in two different ways:

Adding both sides of the two equations, all terms involving d cancel:

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

An alternate form results from re-inserting the substitution: an = a1 + (n 1)d:
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In 499 CE Aryabhata, a prominent mathematician-astronomer from the classical age ofIndian mathematics and Indian astronomy, gave this method in the Aryabhatiya (section2.18) .[1]

So, for example, the sum of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up to the 50th term is

Product

The product of the members of a finite arithmetic progression with an initial element a1,common differences d, and n elements in total is determined in a closed expression

where denotes the rising factorial and denotes the Gamma function. (Note howeverthat the formula is not valid when a1 / d is a negative integer or zero.)

This is a generalization from the fact that the product of the progressionis given by the factorial n! and that the product

for positive integers m and n is given by

Taking the example from above, the product of the terms of the arithmetic progressiongiven by an = 3 + (n-1)(5) up to the 50th term is

Consider an AP

a,(a + d),(a + 2d),.................(a + (n 1)d)

Finding the product of first three terms

a(a + d)(a + 2d) = (a2 + ad)(a + 2d) = a3 + 3a2d + 2ad2

this is of the form

an + nan 1dn 2 + (n 1)an 2dn 1

so the product of n terms of an AP is:

an + nan 1dn 2 + (n 1)an 2dn 1 no solutions
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Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is asequence ofnumbers where each term after the first is found by multiplying the previousone by a fixed non-zero number called the common ratio. For example, the sequence 2, 6,18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... isa geometric sequence with common ratio 1/2. The sum of the terms of a geometricprogression is known as a geometric series.

Thus, the general form of a geometric sequence is

and that of a geometric series is

where r 0 is the common ratio and a is a scale factor, equal to the sequence's startvalue.

Elementary properties

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

Such a geometric sequence also follows the recursive relation

for every integer

Generally, to check whether a given sequence is geometric, one simply checks whethersuccessive entries in the sequence all have the same ratio

The common ratio of a geometric series may be negative, resulting in an alternatingsequence, with numbers switching 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 infinity. 1, the progression is a constant sequence. Between 1 and 1 but not zero, there will beexponential decay towards zero. 1, the progression is an alternating sequence (see alternating series) Less than 1, for the absolute values there isexponential growth towards positive

and negative infinity (due to the alternating sign).

Geometric sequences (with common ratio not equal to 1,1 or 0) showexponential growthor exponential decay, as opposed to the Linear growth (or decline) of an arithmeticprogression such as 4, 15, 26, 37, 48, (with common difference 11). This result wastaken 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
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progression yields a geometric progression, while taking the logarithm of each term in ageometric progression with a positive common ratio yields an arithmetic progression.

Geometric series

A geometric series is the sum of the numbers in a geometric progression:

We can find a simpler formula for this sum by multiplying both sides of the above equationby 1 r, and we'll see that

since all the other terms cancel. Because r 1 for geometric series (r = 1 would give us anarithmetic progression), we can rearrange(for r 1) the above to get the convenientformula for a geometric series:

If one were to begin the sum not from 0, but from a higher term, say m, then

Differentiating this formula with respect to r allows us to arrive at formulae for sums ofthe form

For example:

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

Then

For a series with only odd powers of r
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and

Infinite geometric series

An infinite geometric series is an infinite series whose successive terms have a commonratio. Such a series converges if and only ifthe absolute value of the common ratio is lessthan one ( | r | < 1 ). Its value can then be computed from the finite sum formulae

Since:

Then:

For a series containing only even powers of r,

and for odd powers only,

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

The formulae given above are valid only for | r | < 1. The latter formula is valid in everyBanach algebra, as long as the norm of r is less than one, and also in the field ofp-adicnumbers if | r |p < 1. As in the case for a finite sum, we can differentiate to calculateformulae for related sums. For example,

This formula only works for | r | < 1 as well. From this, it follows that, for | r | < 1,
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Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + is an elementary example of a seriesthat converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is

The inverse of the above series is 1/2 1/4 + 1/8 1/16 + is a simple example of analternating series that converges absolutely.

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sumis

Complex numbers

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

The proof of this comes from the fact that

which is a consequence ofEuler's formula. Substituting this into the original series gives

.

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

[edit] Product

The product of a geometric progression is the product of all terms. If all terms arepositive, then it can be quickly computed by taking the geometric mean of theprogression's first and last term, and raising that mean to the power given by the numberof terms. (This is very similar to the formula for the sum of terms of an arithmeticsequence: take the arithmetic mean of the first and last term and multiply with thenumber of terms.)

(if a,r > 0).

Proof:

Let the product be represented by P:

.
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Now, carrying out the multiplications, we conclude that

.

Applying the sum ofarithmetic series, the expression will yield

.

.

We raise both sides to the second power:

.

Consequently

and

,

which concludes the proof.
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Limit of a sequence

The limit of a sequence (xn) is, intuitively, the unique number or point L (if it exists) suchthat the terms of the sequence become arbitrarily close to L for "large" values of n. If the

limit exists, then we say that the sequence is convergent and that it converges to L.Convergence of sequences is a fundamental notion in mathematical analysis, which hasbeen studied since ancient times.

Formal definition

For a sequence of real numbersA real number L is said to be the limit of the sequence xn, written

if and only if for every real number > 0, there exists a natural number N such that

for every n > N we have | xnL | < .

As a generalization of this, for a sequence of points in a topologicalspace T:

An element L T is said to be a limit of this sequence if and only if for every

neighborhood S of L there is a natural number N such that xn S for all n > N. In

this generality a sequence may admit more than one limit, but if T is a Hausdorff

space, then each sequence has at most one limit. When a unique limit L exists, it

may be written

If a sequence has a limit, we say the sequence is convergent, and that the sequenceconverges to the limit. Otherwise, the sequence is divergent (see also oscillation).

A null sequence is a sequence that converges to 0.

Hyperreal definition

A sequence xn tends to L if for every infinite hypernatural H, the term xH is infinitely closeto L, i.e., the difference xH - L is infinitesimal. Equivalently, L is the standard part of xH

.

Thus, the limit can be defined by the formula

where the limit exists if and only if the righthand side is independent of the choice of aninfinite H.

Comments

The definition means that eventually all elements of the sequence get as close as we wantto the limit. (The condition that the elements become arbitrarily close to all of the

following elements does not, in general, imply the sequence has a limit. See Cauchysequence).
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A sequence of real numbers may tend to or , compare infinite limits. Eventhough this can be written in the form

and

such a sequence is called divergent, unless we explicitly consider it a sequence in the

affinely extended real number system or (in the first case only) the real projective line. Inthe latter cases the sequence has a limit (in the space itself), so could be calledconvergent, but when using this term here, care should be taken that this does not causeconfusion.

The limit of a sequence of points in a topological space T is a special case

of the limit of a function: the domain is in the space with the inducedtopology of the affinely extended real number system, the range is T, and the functionargument n tends to +, which in this space is a limit point of .

Examples

The sequence 1, -1, 1, -1, 1, ... is oscillatory. The series with partial sums 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 +

1/16, ... converges with limit 1. This is an example of an infinite series.

If a is a real number with absolute value | a | < 1, then the sequence an has limit 0.If 0 < a, then the sequence a1/n has limit 1.

Also:

Properties

Consider the following function: f( x ) = xn if n-1 < x n. Then the limit of the sequence ofxn is just the limit of f( x) at infinity.

A function f, defined on a first-countable space, is continuous if and only if it is compatible

with limits in that (f(xn)) converges to f(L) given that (xn) converges to L, i.e.

implies

Note that this equivalence does not hold in general for spaces which are not first-countable.

Compare the basic property (or definition):

f is continuous at x if and only if

A subsequence of the sequence (xn) is a sequence of the form (xa(n)) where the a(n) arenatural numbers with a(n) < a(n+1) for all n. Intuitively, a subsequence omits someelements of the original sequence. A sequence is convergent if and only if all of itssubsequences converge towards the same limit.
http://en.wikipedia.org/wiki/Limit_of_a_function#Limit_of_a_function_at_infinityhttp://en.wikipedia.org/wiki/Affinely_extended_real_number_systemhttp://en.wikipedia.org/wiki/Real_projective_linehttp://en.wikipedia.org/wiki/Limit_of_a_function#Functions_on_topological_spaceshttp://en.wikipedia.org/wiki/Induced_topologyhttp://en.wikipedia.org/wiki/Induced_topologyhttp://en.wikipedia.org/wiki/Affinely_extended_real_number_systemhttp://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/First-countable_spacehttp://en.wikipedia.org/wiki/Continuous_function_%28topology%29http://en.wikipedia.org/wiki/Subsequencehttp://en.wikipedia.org/wiki/Subsequencehttp://en.wikipedia.org/wiki/Continuous_function_%28topology%29http://en.wikipedia.org/wiki/First-countable_spacehttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Limit_of_a_functionhttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Series_%28mathematics%29http://en.wikipedia.org/wiki/Limit_pointhttp://en.wikipedia.org/wiki/Affinely_extended_real_number_systemhttp://en.wikipedia.org/wiki/Induced_topologyhttp://en.wikipedia.org/wiki/Induced_topologyhttp://en.wikipedia.org/wiki/Limit_of_a_function#Functions_on_topological_spaceshttp://en.wikipedia.org/wiki/Real_projective_linehttp://en.wikipedia.org/wiki/Affinely_extended_real_number_systemhttp://en.wikipedia.org/wiki/Limit_of_a_function#Limit_of_a_function_at_infinity


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Every convergent sequence in a metric space is a Cauchy sequence and hence bounded. Abounded monotonic sequence of real numbers is necessarily convergent: this is sometimescalled the fundamental theorem of analysis. More generally, every Cauchy sequence ofreal numbers has a limit, or short: the real numbers are complete.

A sequence of real numbers is convergent if and only if its limit superior and limit inferior

coincide and are both finite.The algebraic operations are everywhere continuous (except for division around zerodivisor); thus, given

and

then

and (if L2 and yn are non-zero)

These rules are also valid for infinite limits using the rules

q + = for q - q = ifq > 0 q = - ifq < 0 q / = 0 ifq

(see extended real number line).History

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involvelimiting processes.

Leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method ofexhaustion, which uses an infinite sequence of approximations to determine an area or avolume. Archimedes succeeded in summing what is now called a geometric series.

Newton dealt with series in his works on Analysis with infinite series (written in 1669,circulated in manuscript, published in 1711), Method of fluxions and infinite series (writtenin 1671, published in English translation in 1736, Latin original published much later) and

Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix tohis Optiks). In the latter work, Newton considers the binomial expansion of (x+o)n which hethen linearizes by taking limits (letting o0).

In the 18th century, mathematicians like Euler succeeded in summing some divergentseries by stopping at the right moment; they did not much care whether a limit existed, aslong as it could be calculated. At the end of the century, Lagrange in his Thorie desfonctions analytiques (1797) opined that the lack of rigour precluded further developmentin calculus. Gauss in his etude ofhypergeometric series (1813) for the first time rigorouslyinvestigated under which conditions a series converged to a limit.

The modern definition of a limit (for any there exists an index N so that ...) was givenindependently by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, little noticedat the time) and by Cauchy in his Cours d'analyse (1821).
http://en.wikipedia.org/wiki/Metric_spacehttp://en.wikipedia.org/wiki/Cauchy_sequencehttp://en.wikipedia.org/wiki/Metric_spacehttp://en.wikipedia.org/wiki/Monotonichttp://en.wikipedia.org/wiki/Completeness_%28topology%29http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferiorhttp://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Zeno_of_Eleahttp://en.wikipedia.org/wiki/Zeno%27s_paradoxeshttp://en.wikipedia.org/wiki/Zeno%27s_paradoxeshttp://en.wikipedia.org/wiki/Leucippushttp://en.wikipedia.org/wiki/Democritushttp://en.wikipedia.org/wiki/Antiphon_%28person%29http://en.wikipedia.org/wiki/Eudoxus_of_Cnidushttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Geometric_serieshttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Joseph_Louis_Lagrangehttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Hypergeometric_serieshttp://en.wikipedia.org/wiki/Bernhard_Bolzanohttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Augustin_Louis_Cauchyhttp://en.wikipedia.org/wiki/Bernhard_Bolzanohttp://en.wikipedia.org/wiki/Hypergeometric_serieshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Joseph_Louis_Lagrangehttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Geometric_serieshttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Eudoxus_of_Cnidushttp://en.wikipedia.org/wiki/Antiphon_%28person%29http://en.wikipedia.org/wiki/Democritushttp://en.wikipedia.org/wiki/Leucippushttp://en.wikipedia.org/wiki/Zeno%27s_paradoxeshttp://en.wikipedia.org/wiki/Zeno%27s_paradoxeshttp://en.wikipedia.org/wiki/Zeno_of_Eleahttp://en.wikipedia.org/wiki/Extended_real_number_linehttp://en.wikipedia.org/wiki/Limit_superior_and_limit_inferiorhttp://en.wikipedia.org/wiki/Completeness_%28topology%29http://en.wikipedia.org/wiki/Monotonichttp://en.wikipedia.org/wiki/Metric_spacehttp://en.wikipedia.org/wiki/Cauchy_sequencehttp://en.wikipedia.org/wiki/Metric_space


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mate sequence

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