# Thread: Functions as power series

1. ## Functions as power series

My real question does not have to do with the problem itself so much as it has to do with a concept within problems like these... or, perhaps, series in general.

$f(x)=\frac{x^3}{(x-2)^2}$

Breaking it down into pieces, starting first with the following...
$\frac{1}{x-2}= \sum_{n=0}^{\infty} \frac{1}{2^{n+1}}x^n$

Now to the next step (where my problems begin)...
$\frac{1}{(x-2)^2}$
$=\frac{d}{dx}\Big[\frac{1}{x-2}\Big]$
$=\frac{d}{dx}\Big[$
$\sum_{n=0}^{\infty} \frac{1}{2^{n+1}}x^n\Big]$

And now my problem...
$=\sum_{n=1}^{\infty}$
$\frac{1}{2^{n+1}}nx^{n-1}$

My problem is in not knowing how we went from starting the sum at n=0 to suddenly starting it at n=1.

Why does this happen and when should it happen?

2. Originally Posted by symstar
My real question does not have to do with the problem itself so much as it has to do with a concept within problems like these... or, perhaps, series in general.

$f(x)=\frac{x^3}{(x-2)^2}$

Breaking it down into pieces, starting first with the following...
$\frac{1}{x-2}= \sum_{n=0}^{\infty} \frac{1}{2^{n+1}}x^n$

Now to the next step (where my problems begin)...
$\frac{1}{(x-2)^2}$
$=\frac{d}{dx}\Big[\frac{1}{x-2}\Big]$
$=\frac{d}{dx}\Big[$
$\sum_{n=0}^{\infty} \frac{1}{2^{n+1}}x^n\Big]$

And now my problem...
$=\sum_{n=1}^{\infty}$
$\frac{1}{2^{n+1}}nx^{n-1}$

My problem is in not knowing how we went from starting the sum at n=0 to suddenly starting it at n=1.

Why does this happen and when should it happen?
Because
$=\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}nx^{n-1}$
and
$=\sum_{n=1}^{\infty}\frac{1}{2^{n+1}}nx^{n-1}$ are the same sum! There is an "n" multiplied in each term. When n= 0, that term is 0.

3. Originally Posted by HallsofIvy
Because
$=\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}nx^{n-1}$
and
$=\sum_{n=1}^{\infty}\frac{1}{2^{n+1}}nx^{n-1}$ are the same sum! There is an "n" multiplied in each term. When n= 0, that term is 0.
Ok, so essentially whenever n=0 makes the term 0, I should make n=1? ...such that:

$=\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}nx^{n-1}$
$=\sum_{n=1}^{\infty}\frac{1}{2^{n+1}}nx^{n-1}$
$=\sum_{n=0}^{\infty}\frac{n+1}{2^{n+2}}nx^n$

Whenever I don't do this, I'm getting the problems wrong.