Does this series converge pointwise or uniformly?

Okay, I'm terribly inexperienced with series convergence, so here is my proof after I read the chapter, please check!

I claim that this series converges pointwise but not uniformly.

Proof.

Define the partial sum

Case 1) If

Then we have

Case 2) If

Then we have

Therefore the series converges pointwise to

I know that I still need to prove this thing is not uniformly convergence, but am I on the right track so far? Thanks!