Is $\sum_{n=2}^\infty\log\left(1+\frac{(-1)^n}{\sqrt{n}}\right)$ convergent? The question is motivated by the following exercise:

Find an example of a sequence of complex numbers $\{a_n\}$ such that $\sum a_n$ converges but $\prod(1+a_n)$ diverges. 

A necessary condition for $\{a_n\}$ is that $\sum|a_n|^2$ must diverges according to the exercise I referred to in a previous question. (This would be also sufficient if the answer to that question is yes.)  
My first question:

Is $$\sum_{n=2}^\infty\log\left(1+\frac{(-1)^n}{\sqrt{n}}\right)$$ convergent? (I don't see a convergence test which would immediately help.)

We would have an example for the exercise above if the answer is no. Can one come up with another example?

$$\log\left(1+\frac{(-1)^n}{\sqrt{n}}\right)=\frac{(-1)^n}{\sqrt{n}}-\frac{1}{2n}+O\left(\frac{1}{n\sqrt{n}}\right)$$
and since the series $\displaystyle\sum_n \frac{(-1)^n}{\sqrt{n}}$ and $\displaystyle\sum_n \frac{1}{n\sqrt{n}}$ are convergent and the series $\displaystyle\sum_n\frac{1}{n}$ is divergent then the given series is divergent.
