real analysis - Show pointwise convergence and (potentially) uniform convergence $sum_{k=1}^inftyfrac{x^k}{k}$

I am looking to show pointwise convergence and (potentially) uniform convergence of the following:

$$\sum_{k=1}^\infty\frac{x^k}{k}$$

I know (from my book) this converges for my given values of $x \in (0,1)$, but I can't figure out how to do this. I tried using the ratio test, but I wasn't able to get an answer that made sense($x$ is what I kept getting). I also tried Weierstrass M-Test, but could only to think to compare it to $$\sum_{k=1}^\infty\frac{1}{k}$$ which doesn't work either. Wolfram says this can be shown using the ratio test.

Can somebody give me an idea of what to use for the M-Test or maybe do the ratio test so I can see if I am doing something wrong?

Edit: I think my pointwise convergence to $x$ is correct, I just want to double-check that this is not uniform convergence. Am I correct?