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Math Help - Combining sums problem

  1. #1
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    Question 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,

    (x-c)^{\text{whatever}} <br />

    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,

    \sum_{n=0}^{\infty} (-1)^{n}2^{n}nx^{n+1} + \sum_{n=0}^{\infty} (-1)^{n}2^{n}nx^{n-1} <br />

    As you can see the,

    (x-c)^{\text{whatever}} <br />

    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,

    n \rightarrow n-1 <br />
,

    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,

    n \rightarrow (n+1) <br />
,

    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!
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  2. #2
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    I suggest writing the first few terms of each and collecting the like terms. See what you come up with...
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