I am trying to determine whether the series $\displaystyle\sum_{n=2}^{+\infty} \ln\left(1-\frac{(-1)^n}{\sqrt n}\right)$ converges or not.
I have tried using a Taylor series of the summand, which gives $$v_n = -\frac{(-1)^n}{\sqrt n} - \frac{1}{2n} + O\left(\frac{1}{n\sqrt n}\right)$$ whose series does not converge because the first term gives an alternating series, the big-O is absolutely convergent but the middle term is the harmonic series, and is thus equivalent to $-\frac{1}{2}\ln n$. Thus the series diverges and the partial sum tends to negative infinity.
However, Wolfram Alpha tells me the exponential of the series, the infinite product $\displaystyle\prod_{n=2}^{+\infty}\left(1-\frac{(-1)^n}{\sqrt n}\right)$ has a nonzero limit, which contradicts that.
Have I been doing something wrong ?
Wolfram Alpha was wrong, and my proof was correct. We indeed have $$ \prod_{n=2}^\infty \left(1-\frac{(-1)^n}{\sqrt n}\right) = 0 $$ as $\displaystyle\sum _{k=2}^n \ln\left(1-\frac{(-1)^k}{\sqrt k}\right) = -\frac{1}{2}\ln(n) + \mathrm O(1)$ as $n $ approaches infinity.