General question about differentiating power series

[Feodalherren]

Say I have a simple series like

$\Sigma^{∞}_{n=0} X^{n}$

When I differentiate this series the first term goes to 0 because it's a constant. Does that mean that I have to adjust the index of the series from n=0 to n=1? If I don't do it, the first term still goes to zero as n(x^(n-1)) when n=0, is 0. My question is, do I even need to bother with the index? It's such a hassle and I'm trying to come up with a plan to save time on my exams. Obviously, if I have to sums and need them together I will change the index.

 

[leehufford]

I think you would have to change the index. We do that in my differential equations class when we find series solutions to DE's. Remember changing the index of summation doesn't change the sum as long as you account for the terms you've stripped out. You just add the stripped out terms to the overall sum. Since zero is the first term, you can omit it. For each positive shift in index another term comes outside the summation.

You might want another opinion first, this is my first time giving advice rather than taking it on this forum.

-Lee

 

[vela]

No, you don't have to change the index as long as when you expand the summation it gives the right result.