<s>[INST] Convergence of the series $1+1/2 -1/3 -1/4 -1/5 +1/6 +1/7 - 1/8 -1/9 - 1/10 + 1/11 + 1/12-...$ I would like to prove that $1+1/2 -1/3 -1/4 -1/5 +1/6 +1/7 - 1/8 -1/9 - 1/10 + 1/11 + 1/12 -...$ converges or diverges
I first showed that $1-1/2 -1/3 +1/4+1/5-1/6-1/7+1/8+1/9-1/10...$ converges by writing it as $1+\Sigma(-1)^n(4n+1)/((2n)(2n+1))$
which converges by A.S.T. However I do not see how to do something similar with this series. The sign has a pattern of 5 and thus I don't think it can be written as an alternating series and thus I do not see how it can converge. I was also unable to find a regrouping which obviously diverges. 
Any help would be appreciated 
 [/INST] Note that you can group all but the first two terms into groups of five terms of the form
$$
-\frac{1}{5k+3}-\frac{1}{5k+4}-\frac{1}{5k+5}+\frac{1}{5k+6} + \frac{1}{5k+7}
< -\frac{1}{5k+3}
$$
This means that the $5k+7$-th partial sum will be bounded above by 
$$
1 + \frac{1}{2} - \frac{1}{3}-\frac{1}{8} - \frac{1}{13} - ... - \frac{1}{5k+3}
$$Hopefully from here you can show that this implies divergence.
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