I have the following series $\displaystyle \sum_{n=1}^{+\infty} \frac{(-1)^{n+1}}{3n + n(-1)^n}$. Does it converge?
I wanted to the alternating series test, but that's not easy because of the two $(-1)^n$, plus it's not monotone decreasing.
I've split it into two series
$n=2n \Rightarrow \sum{-\frac 1 {4m}}\Rightarrow$ diverges,
$n=2n+1 \Rightarrow \sum{\frac 1 {2m}}\Rightarrow$ diverges.
I can't say diverges + diverges = diverges, but I have this hunch, that the positive series diverges faster than the negative series and therefore it diverges. However, I don't know any test I can do to prove that? Alternating, ratio and root fails. I don't know with what I can compare it to. Thanks in advance for your help!