Question

Find the power series of the function:

Answer 1

\[f(x)=\frac{ x ^{3} }{ (x-9)^{2} }\]

Answer 2

ok i'd probably say use this power series\[\frac{1}{1-u}=1+u+u^2+...\]for \(|u|<1\)

Answer 3

I know to use geometric, I just don't know how to turn it into that form.

Answer 4

start with\[\frac{1}{1-\frac{x}{9}}=1+\frac{x}{9}+\frac{x^2}{81}+...\]and then differentiate both sides

Answer 5

for \(|x|<9\)

Answer 6

Why do I start with that?

Answer 7

I understand that you're changing 1/(9-x) to geometric form

Answer 8

because its what u have in denum of first expression\[\frac{1}{1-\frac{x}{9}}\times (-\frac{1}{9})=\frac{1}{x-9}\]

Answer 9

Ok.

Answer 10

But wouldn't you have to multiply it to the top and bottom?

Answer 11

\[\frac{ 1 }{ 1-\frac{ x }{ 9 } }\times \frac{ -1/9 }{ -1/9 }\]

Answer 12

i just wanted to say it can be change to what u have...yes actually u should multiply top and bot

Answer 13

Ok.

Answer 14

So you work from the geometric and then transition it to what you have? You don't try to manipulate the original equation?

Answer 15

I thought we have to somehow manipulate it via deriving or integrating.

Answer 16

thats what came in to my mind first...maybe there is a better way :)

Answer 17

I've been so stuck on this problem for over an hour. :/

Answer 18

ok i hope that helps :) good luck