
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 (xc)^{\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^{n1}
$
As you can see the,
$\displaystyle (xc)^{\text{whatever}}
$
terms are not identical, one is (n+1) and the other is (n1).
So if we wanted to make the two exponents identical for the first sum we would look as,
$\displaystyle n \rightarrow n1
$,
and plug in (n1) 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!

I suggest writing the first few terms of each and collecting the like terms. See what you come up with...