Convergence of a series with alternating denominator - Real Analysis Decide if the series converges absolutely, conditionally, or not at all.
\begin{equation}
\sum_{n=1}^{\infty}\frac{(-1)^n}{(2+(-1)^n)n}
\end{equation}

I'm having a lot of trouble with this one. I know the even terms will be of the form $\frac{1}{3n}$ and the odd terms of the form $-\frac{1}{n}$ but I'm not really sure how I can use that to prove convergence. Any hints would be greatly appreciated.
We have:
$$\sum_{n=1}^{2N+1}\frac{(-1)^n}{(2+(-1)^n)n}=\frac{H_N}{6}-\sum_{k=0}^{N}\frac{1}{2k+1}=\frac{H_N}{6}-H_{2N+1}+\frac{H_N}{2}$$
hence:
$$\sum_{n=1}^{2N+1}\frac{(-1)^n}{(2+(-1)^n)n}=\frac{1}{3}\log N-\log(2N)+O(1)=-\frac{2}{3}\log N+O(1)$$
so the series diverges towards $-\infty$.
I agree with your analysis of the problem:
even terms will be of the form 1/3n and the odd terms of the form −1/n.  Then each of them will converge to the same number 0.  so the whole thing will converge to 0.  Afterward, you can prove by induction.