Useful series for limit comparison test I already know the harmonic series ($1/n$, which diverges) and following the link, $1/n^2$ that converges.
What other useful series could you teach me, or perhaps some general advice? Any website/resource is very welcome also, since most of my search for them reveals only the harmonic one and not much more. (I know how to execute the test!).
Have a nice day!
 A: There is the generalization of what you put above, the so called $p$-series.  Let $p\in \mathbb{R}$, then $$\sum_{n=1}^\infty \frac{1}{n^p}$$ converges if and only if $p>1$.  This can be proven using the integral test mentioned by Shai Covo.
Also of interest are alternating series, such as $$\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}$$ since there are good tests to see if they converge.  (The above sum converges) Specifically, the alternating series test tells us that if we have a sequence $a_n$ with 
(1) $a_n\cdot a_{n+1} <0$ for every $n$ (it alternates signs)
(2) $|a_{n+1}|\leq |a_n|$
(3) $\lim_{n\rightarrow \infty} a_n =0$
Then $\sum_{n=1}^\infty a_n$ converges.  This then brings up the topic of Conditional and Absolute convergence.  For a generalization of the alternating series test, see Dirichlets Test.  (This test allows us to give the conditions of convergence for series such as $\sum_{n=1}^\infty a_n \sin (n)$)
Hope that helps,
A: General advice: consider the integral test for convergence. As an exercise, apply it to the series $\sum\limits_{n = 2}^\infty  {\frac{1}{{n(\log n)^\alpha  }}}$, $\alpha > 0$ fixed. For which $\alpha$ does the series coverge?
Another very useful test is the alternating series test. See also convergence tests.
