**Question Corner and Discussion Area**

Hi!The series you have described is not a geometric series. It is an example of a more general class of series calledMy question is about geometric series. I read about the one that you solved, but this one is a little bit different :

What is the sum from

i= 0 to infinity of (x^i)(i^2)?Thanks.

A key fact about power series is that, if the series converges
on an interval of the form |*x*| < *R*, then it "converges uniformly"
on any closed subinterval of that interval. I won't attempt to explain
what that means, but will mention instead an important consequence
(which is not always true for series that are not power series):
*the series can be integrated and differentiated term by term*,
in the sense that, if you define

then

This means that, if you start with the geometric series which is
known to converge to 1/(1-*x*) when |*x*| < 1 (as described in
the answer to another question), the following is
true for all |*x*| < 1 by differentiating both sides of the equation:

If you multiply both sides by *x* you get something close to what you
want:

Differentiating both sides again and multiplying by *x* again
gives you what you want:

Therefore, your series converges to , provided
|*x*| < 1. (If |*x*| > 1, it diverges).

This particular technique will, of course, work only for this specific example, but the general method for finding a closed-form formula for a power series is to look for a way to obtain it (by differentiation, integration, etc.) from another power series whose sum is already known (such as the geometric series, or a series you can recognize as the Taylor series of a known function).

Most series don't have a closed-form formula, but for those that do, the above general strategy usually helps one to find it.

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