power series section 9.1a. a finite sum of real numbers always produces a real number, but an...

Power SeriesSection 9.1a

A finite sum of real numbers always produces a real number,but an infinite sum of real numbers is not actually a real sum:

Definition: Infinite SeriesAn infinite series is an expression of the form

1 2 3 ,na a a a or1

kk

a

The numbers are the terms of the series;1 2, ,a a is the nth term.na

A finite sum of real numbers always produces a real number,but an infinite sum of real numbers is not actually a real sum:

The partial sums of theseries form a sequence

1 1s a

2 1 2s a a

3 1 2 3s a a a

1

n

n kk

s a

If the sequence of partial sums has a limit S as napproaches infinity, we saythe series converges to S:

11

n kk

a a a S

Otherwise, we say the series diverges.

Guided PracticeDoes the given series diverge?

1 1 1 1 1 1 Can we use the associate property of addition?:

1 1 1 1 1 1 This does NOT work with infinite series!!!

Instead, look at the sequence of partial sums:

1,0,1,0,1,0,1,Since this sequence has no finite limit, then the originalseries has no sum. It diverges.

Guided PracticeDoes the given series diverge?

3 3 3 3

10 100 1000 10n

Look at the sequence of partial sums:

0.3,0.33,0.333,0.3333,This sequence has a limit , which we recognize as 1/3.The series converges to the sum 1/3.

0.3

This last problem is an example of a geometric series,because each term is obtained by multiplying the previousterm by the same constant (1/10 in this case)…

Geometric SeriesThe geometric series

2 3 1 1

1

n n

n

a ar ar ar ar ar

converges to the sum if ,1

a

r1r

and diverges if 1r The interval is the interval of convergence.1 1r

(these are the r values for which the series converges)

Guided PracticeTell whether each series converges or diverges. If itconverges, give its sum.

1

1

132

n

n

The first term is a = 3 and r = 1/2.

The series converges to: 3

1 1 26

11 1 1 1

12 4 8 2

n

The first term is a = 1

and r = –1/2.

The series converges to: 1

1 1 2 2

3

Guided PracticeTell whether each series converges or diverges. If itconverges, give its sum.

2

3

5

k

k

The first term is a = 9/25 and r = 3/5.

The series converges to: 9 25

1 3 59

10

2 3

2 4 8

In this series,

The series diverges.2 1r

Representing Functions by SeriesIf , then the geometric series formula gives:1x

2 3 11

1nx x x x

x

The expression on the rightdefines a function whosedomain is the set of all realnumbers .1x

The expression on the leftdefines a function whosedomain is the interval ofconvergence, .1x

The equality is understood to hold only in this latter domain,where both sides of the equation are defined. On thisdomain, the series represents the function 1/(1 – x).

Representing Functions by SeriesIf , then the geometric series formula gives:1x

2 3 11

1nx x x x

x

Let’s graph the function together with some partial sums ofthe series. Graph in [–4.7, 4.7] by [–2, 4]:

1

1

1y

x

2

2 1y x x 2 3

3 1y x x x 2 3 4

4 1y x x x x

The graphs behave similarly, but onlyon the interval of convergence!!!

Definition: Power SeriesAn expression of the form

20 1 2

0

n nn n

n

c x c c x c x c x

is a power series centered at x = 0. An expression of the form

20 1 20

n

nn

c x a c c x a c x a

nnc x a

is a power series centered at x = a. The term isthe nth term; the number a is the center.

nnc x a

Guided PracticeGiven that 1/(1 – x) is represented by the power series

21 nx x x on the interval (–1, 1), find a power series that represents

1 1 x on 1,11. 2 31

nx x x x

1x x on 1,12. 2 3 4 11

n nx x x x x

1 1 2x on 1 2,1 23. 2 31 2 4 8 2

nx x x x

Guided PracticeGiven that 1/(1 – x) is represented by the power series

21 nx x x on the interval (–1, 1), find a power series that represents

1 1

1 1x x

and give its interval of convergence4.

2 31 1 1 1x x x

1 1n nx

This geometric series converges for: 1 1 1x 0 2x

Interval of convergence: 0,2

Guided PracticeGiven that 1/(1 – x) is represented by the power series

21 nx x x on the interval (–1, 1), find a power series that represents

1 1 1

3 3 1 1x x

and give its intervalof convergence

5.

2 31 1 1 11 1 1

3 3 3 3x x x

11

3

nn

x

This geometric seriesconverges for: 1 1 1x

0 2x Interval of

convergence: 0,2