I have to investigate convergence of series $$\sum_{k=10}^{+\infty}\frac{(-1)^k}{k+(-1)^k}$$ It certainly does not converge absolutely, because it is basically a harmonic series with every two elements flipped and if harmonic series converged, or greater than series $$\sum_{k=10}^{+\infty}\frac{1}{k-1}$$ However, I do not know, how to prove convergence of this series overall. I thought about the fact, that it is basically a harmonic series(well, alternating harmonic or whatever I should call it) with every two elements flipped and if normal "alternating harmonic series" converges, then this rearangement could technically converge too. However, I am not sure if this is sufficient enough and if I can do this and I am afraid that it is not. I was considering those theorems that say "every divergent series with alternating elements can be rearranged to make arbitrary finite sum" and I am not sure if it holds with convergent series, respectively, if it holds against convergent series and infinity - that you can rearrange convergent series to make divergent...
Neverthless, there should be an easy way to prove that this sum is convergent/divergent, since it is in our textbook.
Also I thought, since it is certainly lesser or equal to $1/(k-1)$ and greater or equal to $1/(k+1)$ assigning numbers $S_1$ and $S_2$ to sums of these "alternating harmonics" and just saying that it is lower bounded by $S_2$ and upper bounded by $S_1$, since those series are convergent and have to be upper and lower-bounded too. But also, I am not sure if this is enough.