Differentiating under the summation I saw on the Wikipedia page for differentiation under the integral that it could also be applied to summations. Here is the link:
http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign#Applications_to_series
It says exactly that "Differentiating under the integral can also be applied to differentiating under summation, interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence." 
Does this mean that I can use differentiation under the summation to evaluate a series? If so, can someone give an example for me to see? 

You can differentiate under summation sign, provided the series converges "nicely".
An example is the following:
$\dfrac{1}{1-x}=1+x+x^2+\cdots$.
Then $\dfrac{1}{(1-x)^2}=1+2x+3x^2+\cdots$
The things to notice are the following:
1) The formula for $\dfrac{1}{1-x}$ only holds for $|x|<1$.
2) The series converges uniformly on $[-R,R]$ for any $0<R <1$
3) The formula for $\dfrac{1}{(1-x)^2}$ holds in the same range of values of $x$, i.e. $|x| \leq 1$.
In general, if a series converges uniformly, you can differentiate under the summation sign, and get a series that has radius of convergence at least as large as the given series. 
