TITLE: Convergence of Series
QUESTION [6 upvotes]: At university, we are currently introduced in various methods of proving that a series converges. For example the ComparisonTest, the RatioTest or the RootTest. However we aren't told of how to calculate the point of convergence. The only series of which we know the point of convergence is the geometric series. 
I believe this to be a non trivial task in general, but there have to be at least some methods to calculate the point of convergence for simple series, haven't there?
If you could point me in the direction of some of these methods I would very much appreciate it.
Thanks in advance

REPLY [7 votes]: The short answer is that there are no general methods and we use whatever's available.  The first thing you can do is learn the Taylor series of common functions; then evaluation of the Taylor series at a point where the series converges to the function corresponds to evaluation of the function.  For example, $e^x = \sum \frac{x^n}{n!}$, hence $e = \sum \frac{1}{n!}$.  Less trivially, $\ln (1 + x) = \sum \frac{(-1)^{n-1} x^n}{n}$, hence
$$\ln(2) = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} \pm ....$$
The more Taylor series you know, the more series you can evaluate "explicitly" in some sense.  
Beyond that, life gets difficult.  Mastering the use of Taylor series is already highly nontrivial - especially recognizing when the method is applicable - so worry about learning how to do that properly first.