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? 

Hint. You may use the ratio test, evaluating
$$
\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}(x)}{a_n(x)}\right|
$$ with $$ a_n(x)=\frac{x^n}n.$$
