SECT. 9-3 POWER SERIES

Sect. 9-8 in textbook

Power Series

A series with variable terms like

Is called a power series. Note that this series is a geometric power series. If it converges, the variable x must be

......1 32 nxxxx

11 x

The Interval of Convergence

11 x

Power Series

1. For these x-values of this Geometric Power series

This means for these x-values, the function

can be written as

......1 32 nxxxx

xxf

1

1)(

0

)(n

nxxf

Power Series General Form

A power series is a series of the form

where x is a variable and the cn’s are constants called the coefficients of the series.

...0

33

2210

n

nn xcxcxccxc

Find a power series for the following function. Show four terms and a general term. Also give the series using sigma notation and give the interval of convergence

xxf

1

1)( )2

Find a power series for the following function. Show four terms and a general term. Also give the series using sigma notation and give the interval of convergence

xxf

21

3)( )3

Find a power series for the following function. Show four terms and a general term. Also give the series using sigma notation and give the interval of convergence

xxf

3

1)( )4

Power Series By Substitution 5. Using the power series from example 2 make a new power series for )( 2xf

Power Series By Substitution 6. Using the power series from example 3 make a new power series for xg

Power Series By Differentiation 7. Use the power series for

to write a power series for

xxf

1

1)(

)(' xf

...1)( 32 xxxxf

Power Series By Differentiation 8. If

Find )(' xj

...!

...!4!3!2

1)(432

n

xxxxxxj

n

9. Identify the function j(x)

Power Series By Integration 10. Since

then

2 311 1

1n nt t t t

t

2 3

0 0

11 1

1

x x n ndt t t t t dtt

Power Series By Substitution 11. Use the result of example 10 to write a power series for )ln()( xxf

... 32

)1ln(32

xx

xx

Power Series By Substitution 12. Write a power series for )ln()( 2 xxxg

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