Regarding sequences of functions $(f_n(x))$, I can wrap my head around the idea that uniform convergence $\Rightarrow$ pointwise convergence, but pointwise convergence does not imply uniform convergence.
However, regarding series of functions $\sum f_n(x)$, I am not sure if I totally understand. I have proved that if a series converges uniformly on $\mathbb R$, then it is pointwise convergence on $\mathbb R$ and $\sup |f_n(x)|$ converges to 0 as n goes to infinity. But I am not sure of the converse. I was pretty sure the converse did not hold true, but I cannot find a counter example.
Is it similar to sequences and the converse does not hold true? Or am I thinking of it wrong and there is a way to prove that it does also hold true?
The series $\sum_{n=0}^\infty x^n$ converges pointwise to $\frac1{1-x}$ on $(-1,1)$. However, the convergence is not uniform, since each function $\sum_{n=0}^N x^n$ is bounded on $(-1,1)$, but $x\mapsto\frac1{1-x}$ isn't.