11.8 Power Series

Power Series

We now consider infinite series of the form:

0

)(n

nn axc

x is a variable

a is a constant

c’s are coefficientsthat depend on n

Example

0n

nx Note here that a = 0 and cn = 1 (for each n)

For what values of x doesthe series converge?

Need |x|<1

Example

0n

nx Note here that a = 0 and cn = 1 (for each n)

If the series converges,what is the sum?

Sum = 1/(1-x)

Example

We just showed that

xxxxx

n

n

1

1.....1 32

0

WOW! This function can be represented as an infinite series

If |x|<1

Example

For what values of x do the following series converge?

1

1

.)

1

)1(.)

n

nn

n

nn

xnb

n

xa

General case

0

)(n

nn axc

For a given power series there are three possibilities:

1.) The series converges for all values of x

2.) The series only converges for x=a

3.) There is a positive number R, called the radius of convergence, such that the series

converges if |x-a|<R and diverges if |x-a|>R

(R=infinity)

(R=0)

Radius of Convergence

• Use ratio or root test to find R

• At the endpoints x=a+R and x=a-R, anything can happen! The series may converge or diverge…further testing must be done!

Why do we care???

• We can represent functions as infinite power series (11.9 functions as power series)

• Note that a power series is an infinite polynomial defined by the coefficients cn

• We like polynomials – Easy to integrate and differentiate

Closer look at

1

1

1.....1)( 32

0

x

xxxxxxf

n

n

Can we use this toexpress other functions as

power series?

Example

Express

as a power series.

41

3)(

xxf

Coming Soon!

• What about other functions such as

?)cos()(

?)sin()(

?)(

xxf

xxf

exf x

Can we express theseas power series???