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,