Differentiating geometric series On Wikipedia it is stated that by differentiating the following formula holds:
$$ \sum_n n q^n = {1\over (1-q)^2}$$
Does this not require a proof? It seems to me because the series is infinite it is not clear that differentiation commutes with taking the limit. 

How to prove this?

 A: It does require proof.  That the derivative of a sum of finitely many terms is the sum of the derivatives is proved in first-semester calculus, but it doesn't always work for infinite series.  For example, let
$$
g_n(x) = \frac{\sin(nx)} {n^2}.
$$
Then $\displaystyle\sum_{n=1}^\infty g_n(x) \vphantom{\dfrac \sum {\displaystyle \int}}$ converges for every value of $x$ (since the absolute value of each term is $\le 1/n^2$ and $\sum_n 1/n^2 < \infty$). And
$\displaystyle \sum_{n=1}^\infty g_n'(x) = \sum_{n=1}^\infty \frac{\cos(nx)}{n}$, and that diverges when $x=0$.
However, every power series converges uniformly on sets bounded away from the boundaries of the interval of convergence.  I.e.
$$
f(x) = \sum_{n=0}^\infty a_n (x - c)^n
$$
converges for $x$ in some interval $(c-r,c+r)$.  (In some cases it also converges at one or both of the endpoints.  The number $r$ is the radius of convergence.  A set that is "bounded away from the endpoints" is any subset of an interval of the form $(c-r+\varepsilon,c+r-\varepsilon)$.  The series will fail to converge uniformly on $(c-r,c+r)$ if there is a vertical asymptote at either endpoint, but it converges pointwise on that set and uniformly on every set bounded away from the endpoints (i.e. no matter how small $\varepsilon$ above is).  At this point I'll cite a book, but maybe I'll be back later. Walter Rudin's Principles of Mathematical Analysis, third edition, page 173.  You'll find a proof of term-by-term differentiability of power series in $(c-r,c+r)$, using uniform convergence.
A: I think that since the series is absolutely convergent, taking a limit and differentiating commute
A: Power series are termwise differentiable in their open interval of convergence. This is a basic property of power series.
