infinite sequences and summation notation

Infinite Sequencesand Summation Notation

An arbitrary infinite sequence maybe denoted as follows.

a1, a2, a3, a4, ... an, ...

Each number ak is a term of the sequence.

The sequence is ordered in thesense that there is a first term a1,a second term a2, a thirty third terma33, and if n represents an arbitrarypositive integer, and nth term an.

For our work we will restrict the domain to the set of positive integers and define an infinite sequence as:

An infinite sequence is a function whose domain is the set of positiveintegers.

For our work the range of aninfinite sequence will be the set of real numbers.

With these definitions each positiveinteger n in the domain of the sequence corresponds to a real number ƒ(n) = an.

Example 1. Find the first four termsand the tenth term of each sequence.

b. {2 + (0.1)n }

a. nn+1

c. (-1)n+1 n2

3n-1

d. {4}

To find the above replace n with 1, 2, 3, 4, and then 10.


a. nn+1

11+1 =

1 2

22+1 =

2 3

33+1 =

3 4

44+1 =

4 5

1010+1 =

10 11


b. {2 + (0.1)n } 2 + (0.1)1 = 2.12 + (0.1)2 = 2.012 + (0.1)3 = 2.0012 + (0.1)4 = 2.0001

2 + (0.1)10 = 2.0000000001


c. (-1)n+1 n2

3n-1

(-1)1+1 12

3•1-1= 1

2

(-1)2+1 22

3•2-1= -4

5


c. (-1)n+1 n2

3n-1

(-1)3+1 32

3•3-1= 9

8

(-1)4+1 42

3•4-1= -16

11


c. (-1)n+1 n2

3n-1

(-1)10+1 102

3•10-1= -100

29


d. {4 } 4 = 44 = 44 = 44 = 44 = 4

Advanced Algebraand Trigonometry

Assignment 04-01-03 NBP#40

Page 671 Exercises 1 - 15oddCopy each problem.Show your work.

Underline your answer.


Notes04-02-03

NBP#41Infinite Sequences

and Summation NotationPages 664 - 670

Definition: Recursive sequence

The first term a1 is stated along with a rule for obtaining any term ak+1 from the preceding term ak whenever k > 1.

Example 2. Find the first four termsand the nth term of the infinite sequence defined recursively as:

a1 = 3 ak+1 = 2ak for k > 1.

a2 = 2•3

a3 = 2•2•3

a4 = 2•2•2•3 an = 2n-1•3

We sometimes need to find the sum of many terms in an infinite sequence. To express these sumseasily we use summation notation.

Given an infinite sequencea1, a2, a3, ... , an, ...

Given an infinite sequencea1, a2, a3, ... , an, ...

The symbol ak represents the sum of the first m terms.

m

k=1

ak = a1 + a2 + a3 + ... + am

m

k=1

The symbol ak represents the sum of the first m terms.

m

k=1

ak = a1 + a2 + a3 + ... + am

m

k=1

The symbol represents a sum, ak represents the kth term.

ak = a1 + a2 + a3 + ... + am

m

k=1

The symbol represents a sum, ak represents the kth term.

The letter k is the index of summation, or the summationvariable. 1 and m indicate the smallest and largest values of thesummation variable.

Example 3. Evaluate a sum.

∑ k2(k-3) 4

k=1

∑ k2(k-3) 4

k=1= 12(1-3) + 22(2-3)

+ 32(3-3) + 42(4-3)

= (-2) + (-4) + 0 + 16= 10


Assignment 04-01-03 NBP#42

Page 671 Exercises 17 - 31oddCopy each problem.Show your work.

Underline your answer.


Notes04-03-03

NBP#43Infinite Sequences

and Summation NotationPages 664 - 670

We are talking about Sn.For example given the infinite sequencea1, a2, a3, ... , an, ...S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3

S4 = a1 + a2 + a3 + a4

In general Sn = ∑ ak = a1+a2+...+an

n

k=1

We may also writeS1 = a1

S2 = S1 + a2

S3 = S2 + a3

S4 = S3 + a4

Sn = Sn-1 + an

The real number Sn is called the nth partial sum of the infinite sequence a1, a2, a3, ... an, ...

The sequence S1, S2, S3, ... Sn, ...is called a sequence of partial sums.

Example 4. Finding the terms of a sequence of partial sums.

Find the first four terms and the nthterm of the sequence of partial sumsassociated with the sequence1, 2, 3, ..., n, ... of positive integers.

The first four terms are:S1 = a1 = 1

Example 4. Finding the terms of a sequence of partial sums.The first four terms are:S1 = a1 = 1

S2 = S1 + a2 = 1 + 2 = 3

S3 = S2 + a3 = 3 + 3 = 1+2+3 = 6

S4 = S3 + a4 = 6 + 4 = 1+2+3+4 = 10


The nth partial sum Sn can be written in either of the followingforms.

Sn = 1+2+3+...+(n-2)+(n-1)+n

Sn = n+(n-1)+(n-2)+...+3+2+1


Sn = 1+2+3+...+(n-2)+(n-1)+n

Sn = n+(n-1)+(n-2)+...+3+2+1

When I add these two equations

2Sn = (n+1)+(n+1)+...+(n+1)


2Sn = (n+1)+(n+1)+...+(n+1)

Since the term (n+1) appear n times on the right side of the equations we get.

2Sn = n(n+1) or Sn =n(n+1) 2

Sum of a constantIf ak is the same for every positiveinteger k, say ak = c for a realnumber c, then.

= c + c + c + ... + c

= nc

∑ ak = n

k=1a1 + a2 + a3 + ... + an

Sum of a constant Part 2If ak is the same for every positiveinteger k, say ak = c for a realnumber c, then.

= nc - (m-1)c

∑ c = n

k=m∑ c - n

k=1∑ c m-1

k=1

= (n - m + 1)c

Sum of a constant - Illustrations

4 • 7∑ 7 = 4

k=1

∑ π = 10

k=1

∑ 9 = 8

k=3(8 - 3 + 1)9

= 28

10 • π = 10π

= (6)9 = 54

Sum of a constant - Illustrations

∑ 5 = 20

k=10(20 - 10 + 1)5 = (11)9 = 55

Sn = ∑ ak = a0+a1+a2+...+an

n

k=0

If the first term of an infinite sequence is a0, as in a0, a1, a2, ... , an, ...

which is the sum of the first n+1terms of the sequence.

For example ∑ ak = n•ak

n

j=1

If the summation variable does not appear in the term ak, then the entireterm must be considered a constant.

since j does not appear in theterm ak.

Theorem on sums.

n

k=1∑ (ak + bk)

If a1, a2, … , an, … and b1, b2, … , bn, … are infinite sequences, forevery positive integer n,

n

k=1∑ (ak

n

k=1∑ bk)= +

Theorem on sums.

n

k=1∑ (ak - bk)


n

k=1∑ (ak

n

k=1∑ bk)= -

Theorem on sums.

n

k=1∑ cak


n

k=1c(∑ ak) =

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