TITLE: Does this sum converge or not?
QUESTION [5 upvotes]: I have the following sum:$$1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}-\frac{1}{9}-\frac{1}{10}+++++------...$$
How can I get an expression for its partial sum ${s_n}$? 
I know that I can´t rearrange terms unless it converges absolutely, so which is the best way to approach this kind of series?
Any help will be appreciated.

REPLY [3 votes]: What you can do is group together sequences of consecutive terms if they have the same sign.
A typical group of consecutive, equally signed terms is:
$$\pm\sum_{j=t(n)+1}^{t(n+1)}\frac1j$$
where $t(n)$ is the $n$th triangular number, that is, $t(n)=1+2+\cdots+n=\frac{n(n+1)}2$.
and now we can try to bound this sum:
$$0<\sum_{j=t(n)+1}^{t(n+1)}\frac1j\le\frac{n+1}{t(n)+1}=\frac{n+1}{1+\frac{n(n+1)}2}\stackrel{n\to\infty}\to0$$
Since it is an alternating series whose terms tend to $0$, the sum converges.