Question

Use differentiation to find a power series representation for: a.) 1/(1+x)^2 b.) use part a. to find a power series for: 1/(1+x)^3.

I can do the first part, but I cant figure out the second part. I am using the stewart calc book so I cant figure it out by the examples, and the solutions manual skips steps. Please help!!!

Answer 1

Pix of both problem and solution.

Answer 2

is it cool to pm him?

Answer 3

Of course! Zarkon you have any ideas??

Answer 4

yes

Answer 5

not sure how the -1/2 got in front of the d/dx

Answer 6

So from part a you get that:
\[f(x)= \sum_{0}^{\infty} (-1)^n (n+1)x^n \]

Answer 7

Then you take the derivative of the series.

Answer 8

and the derivative of the function..

Answer 9

These two are still equal and on the right hand side you have a function with 1/(1+x)^3 in it

Answer 10

left*

Answer 11

ok, take d/dx of series u posted, equal it to 1/(1+x)^3?

Answer 12

no you can rearrage the equation so just that is on the left hand side

Answer 13

here check this out:

Answer 14

All they did was differentiate both sides, then divide both sides by -2

Answer 15

Then change your indexes so that it is a valid power series and we have x^n

Answer 16

thats where I am lost where did the -2 come from?

Answer 17

nvmn

Answer 18

=P

Answer 19

i see it now.

Answer 20

damn ok thanks for the help. got final tomorrow and been at this all day. think its time for a break.

Answer 21

damn good luck!