<s>[INST] How to evaluate $\sum_{n=1}^{\infty}a_n$? If 
$$a_{n}=1-\frac{1}{2}+\frac{1}{3}-\cdots +\frac{\left ( -1 \right )^{n-1}}{n}-\ln 2$$
then how to eveluate 
$$\sum_{n=1}^{\infty}a_n$$
does it converge?
 [/INST] By using Taylor expansion, we have
\begin{align}
a_{n}=(-1)^{n+1}\left(\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\cdots\right)
\end{align}
which alternates signs and converges to zero (note that $|a_{n}|<\frac{1}{n+1}$.) Also, 
\begin{align}
|a_{n}|&=\left(\frac{1}{n+1}-\frac{1}{n+2}\right)+\left(\frac{1}{n+3}-\frac{1}{n+4}\right)+\cdots \\
&>\left(\frac{1}{n+2}-\frac{1}{n+3}\right)+\left(\frac{1}{n+4}-\frac{1}{n+5}\right)+\cdots=|a_{n+1}|
\end{align}
so the series converges by alternating series test. 
But it seems to be difficult to find closed form.
</s>