Question

(a) Use differentiation to find a power series representation for

f(x) =

1
(5 + x)2
.

f(x) =

?
sum.gif
n = 0


What is the radius of convergence, R?
R =

(b) Use part (a) to find a power series for

f(x) =

1
(5 + x)3

Answer 1

Answer:1/(3+x)=(1/3)

1

(1−(−x/3))

1/(3+x)=(1/3)1(1−(−x/3))

Use this fact that

1/(1−t)=

∑

∞

0

t

n

(∗)

1/(1−t)=∑0∞tn (∗)

where

|t|<1

|t|<1

. I mean take

t=−x/3

t=−x/3

and...

Note that

|−t|=|t|<1

|−t|=|t|<1

is the radius of convergence

Explanation: