Power Series Expansion Homework: Multiplication & n-k Addition Method

[tak13]

Homework Statement

I am doing this multiplication with power series and I am just stuck at this one and other questions that similar to this one.
http://img5.imageshack.us/img5/9526/img1261r.jpg

Homework Equations

The Attempt at a Solution

It seems that I suppose to add n-k wherever I see a "n" but it doesn't seem right.
The highlighted part is the part where I stuck.
The answer for this problem is the one that I circled.

 

[Sethric]

There should be a double sum there, with a sum over k from k = 0 to n. Then you should be able to just solve what that sum would be.

 

[tak13]

Ah yes, I know the part I highlighted should have the Sum of N parenthesis then the Sum of K but I don't know how they get to that answer.

 

[SammyS]

...

The Attempt at a Solution

It seems that I'm supposed to add n-k wherever I see a "n" but it doesn't seem right.
The highlighted part is the part where I stuck.
The answer for this problem is the one that I circled.

What do you get if you multiply the first few terms of f(x) times the first g(x) ?

(fg)(x) = (1 + 2(x-2) + 3(x-2)² + 4(x-2)³ + ... ) (1 + (x-2) + (x-2)² + (x-2)³ + ...)

What is the "k" in your sum?

 

[LCKurtz]

You are using the Cauchy product:

$$(\sum_{n=0}^\infty a_nx^n)(\sum_{n=0}^\infty b_nx^n)=(\sum_{n=0}^\infty c_nx^n)$$

where

$$c_n = \sum_{k=0}^n a_kb_{n-k}$$

In your case a_k = k+1 and b_k = 1. Figure out what you get for c_n.