Testing for convergence in a series

Question

Looking at the following power series,

$$\sum _{n=1}^{\infty} \frac{x^n}{n} = x+\frac{x^2}{2}+\frac{x^3}{3}...$$

I am trying to find the values of $x$ that make the series converge, and what function it converges to.

My thinking was to use the ratio test,

$$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}| = \lim_{n\to\infty}|\frac {x^{n+1}}{x^n}|= x$$

For $|x|<1$ the series converges by the ratio test.
For $|x|>1$ the series diverges by the tatio test.

But what about $|x|=1$, how do I check if this converges or not? By the ratio test this in inconclusive

Answer 1

For $x = -1$, alternating series test.

For $x=+1$ it's the harmonic series.

Answer 2

We may conclude through the Dirichlet test. Let $x=e^{i\theta},\theta\in[0,2\pi)$. We can then see that for $\theta\ne0$, then $\sum_{n=1}^Nx^n$ is bounded (geometric series) and since $\frac1n$ approaches $0$ monotonically, then $\sum_{n=1}^\infty\frac{x^n}n$ converges.