Convergence of the power series I like to determine where the following power series converges.
$$\sum_{k=1}^\infty \dfrac{x^k}{k}  $$
Since the harmonic series diverges, I think that the series would converge if I make the numerator small by forcing $|x|<1$, but I cannot rigoroulsy show where the series converges.
How should I approach this problem? 

To see that the series converges for $|x|<1$ note that 
$$\sum_{k= 1}^\infty \frac{|x|^k}{k} \leq \sum_{k= 1}^\infty |x|^k < \infty$$
this means that $\forall \epsilon >0$ $\exists\, N \in \Bbb{N}$ such that
$$m,n> N \Rightarrow \bigg|\sum_{k = n}^m  \frac{x^k}{k} \bigg|\leq \sum_{k= 1}^\infty |x|^k < \epsilon$$
So the series converges
