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Finding the Sum of a Power Series

Asked by Khanh Son Lam, student, College de Maisonneuve on January 24, 1998:
Hi!

My 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.

The series you have described is not a geometric series. It is an example of a more general class of series called power series, which are of the form
```        inf
-            n
\       a   x
/        n
-
n=0
```
where the coefficients a_n don't depend on the variable x. In your example, a_n = n^2.

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

```               inf
-          n
f(x) = \     a   x ,
/      n
-
n=0
```

then

```                inf
-          n-1
f'(x) = \   n a   x     .
/      n
-
n=0
```

```        inf
-    n
\   x
/
-
n=0
```
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:
```                 inf
1      -       n-1
-----  = \   n  x     .
2   /
(1-x)    -
n=0
```

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

```                 inf
x      -       n
-----  = \   n  x .
2   /
(1-x)    -
n=0
```

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

```                  inf
1 + x    -    2   n
x -----  = \   n  x .
3   /
(1-x)    -
n=0
```

Therefore, your series converges to (x+x^2)/(1-x)^3, 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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