I'm trying to find out for which $|z|=1$ the series $$\sum_{n=1}^\infty{}\frac{z^n}{n}$$ converges. It diverges for $z=1$ (harmonic series) and converges for $z=-1$ (alternating harmonic series). I now know that this is the Taylor series for $-log(1-z)$ which doesn't really help me (although it was interesting to find out)?
I found proofs that the series converges for all $|z|=1$ except for this single $z=1$, but we're asked to proof convergence without using test methods we didn't learn (Dirichlet's test, Abel's test) and therefore are not allowed to use. Have you got any hints on how to get to the wanted result more "heuristically"?
Root criteria gives you: $$\sqrt[n]{\lvert \frac{z^n}{n}\rvert} = t $$ with $ \lim\limits_{n \to \infty} \sqrt[n]n = 1$ you get $\lvert z \rvert = t$
Root criteria now says if $t \lt 1$, the series converges and for $t \gt 1$ the series diverges. For $t=1$ the root criteria does not apply, but for $t=1$ only the cases $z=1$ and $z=-1$ exist for which you already know what happens