# Combining sums problem

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• Oct 3rd 2010, 06:20 PM
jegues
Combining sums problem
The problem statement, all variables and given/known data
I'm having some confusion about combining sums. Our goal when combining these sums is to have the,

$\displaystyle (x-c)^{\text{whatever}}$

term to be the same in both sums.

My confusion is better explained in an example. (see below)

The attempt at a solution

Let's say we have the following 2 sums and we want to simplify them into one sum,

$\displaystyle \sum_{n=0}^{\infty} (-1)^{n}2^{n}nx^{n+1} + \sum_{n=0}^{\infty} (-1)^{n}2^{n}nx^{n-1}$

As you can see the,

$\displaystyle (x-c)^{\text{whatever}}$

terms are not identical, one is (n+1) and the other is (n-1).

So if we wanted to make the two exponents identical for the first sum we would look as,

$\displaystyle n \rightarrow n-1$,

and plug in (n-1) where all the n's used to be in the first sum, and change the starting point of the sum to 1

Now for the second sum, we would look as,

$\displaystyle n \rightarrow (n+1)$,

and plug in (n+1) where all the n's used to be in the second sum,

***Here's where I get confused***

But my professor had mentioned to the class that this would not change the starting point of the sum to n= -1, it stays at n=0.

Why is that? Can someone please clarify?

Thanks again!
• Oct 3rd 2010, 08:09 PM
Prove It
I suggest writing the first few terms of each and collecting the like terms. See what you come up with...