Investigate convergence of the following series Investigate the corvergence of the following series?
$1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+\frac{1}{13}+\frac{1}{15}-\frac{1}{8}+ \ldots$
$1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}-\frac{1}{8}-\frac{1}{9}-\frac{1}{10}+\frac{1}{11}+\frac{1}{12}-\frac{1}{13}-\frac{1}{14}-\frac{1}{15}+ \ldots$
I need a hit about which cirterion to use in each of those cases and i do not know how to find a general term of those twos sequence?

$$\begin{eqnarray*}S_1&=&\sum_{n\geq 0}\left(\frac{1}{4n+1}+\frac{1}{4n+3}-\frac{2}{4n+4}\right)=\sum_{n\geq 0}\int_{0}^{1}\left(x^{4n}+x^{4n+2}-2 x^{4n+3}\right)\,dx\\&=&\int_{0}^{1}\frac{1+x^2-2x^3}{1-x^4}\,dx=\int_{0}^{1}\frac{dx}{1+x}+\int_{0}^{1}\frac{x\,dx}{1+x^2}=\log 2+\frac{1}{2}\log 2\\&=&\color{red}{\frac{3}{2}\log 2}.\end{eqnarray*}$$
In general, the Dirichlet's criterion gives that, if $\{a_n\}$ is a sequence with bounded partial sums and $\{b_n\}$ is a decreasing sequence converging to zero, then the series $\sum_{n\geq 1}a_n b_n$ converges (proof: partial summation). In the first case, if we take $\{a_n\}=\{1,1,0,-2,1,1,0,-2,\ldots\}$ and $b_n=\frac{1}{n}$ we have that $S_1$ is convergent.
In the second case, if we take $\{a_n\}=\{1,1,-1,-1,0,1,1,-1,-1,0\}$ and $b_n=\frac{1}{n}$ we have that
$$ T_2 = \sum_{n\geq 0}\left(\frac{1}{5n+1}+\frac{1}{5n+2}-\frac{1}{5n+3}-\frac{1}{5n+4}\right)$$
converges to some constant (for instance, $\frac{\pi}{5}\sqrt{2+\frac{2}{\sqrt{5}}}=1.068959$ by the previous method), hence
$$ S_2 = \sum_{n\geq 0}\left(\frac{1}{5n+1}+\frac{1}{5n+2}-\frac{1}{5n+3}-\frac{1}{5n+4}+\frac{1}{5n+5}\right)$$
diverges.
