As the title says i have to find out whether the series converges.
I know that every term can be represented by $a_n=(-1)^{n+1}\frac{n+1}{2n}=(-1)^{n+1}\frac{1}{2}(1+\frac{1}{n})$. Now we take the partial sum $s_k=\sum(-1)^{n+1}\frac{1}{2}+\sum{(-1)^{n+1}\frac{1}{n}}$ We now take the limit $\lim{s_k}$ and we know the first thing diverges($\sum{(-1)^{n+1}\frac{1}{2}}$) so the whole thing diverges. Is that correct?
Unfortunately, that's not correct; you can't split up the terms on a series that might have conditional convergence and expect convergence status to be maintained.
Instead, argue from first principles: for a series $\sum_n a_n$ to be convergent of whatever sort, the terms $a_n$ certainly have to go to zero (why?). Do these?