Convergence of the series $\sum_{n\geq 2}\text{log}(1+\frac{(-1)^n}{\sqrt{n}})$

Question

How can I determine the convergence and absolute convergence of $$\sum_{n\geq 2}\text{log}\left(1+\frac{(-1)^n}{\sqrt{n}}\right)$$ I have seen that the series $$\sum_{n\geq 1}\frac{(-1)^n}{\sqrt{n}}$$ converges (not absolutely).

Answer 1

As Jean-Claude Colette commented,

$\begin{array}\\ s(m) &=\sum_{n=2}^m\log\left(1+\frac{(-1)^n}{\sqrt{n}}\right)\\ &=\sum_{n=2}^m\left(\frac{(-1)^n}{\sqrt{n}}+\frac{1}{2n}+O(\frac1{n^{3/2}})\right)\\ &=\sum_{n=2}^m\frac{(-1)^n}{\sqrt{n}}+\sum_{n=2}^m\frac{1}{2n}+\sum_{n=2}^mO(\frac1{n^{3/2}})\\ \end{array} $

and the first and third sums converge and the second sum diverges so the sum diverges both conditionally and absolutely.