Example of a series of functions that converges absolutely but does not converges uniformly?

Question

Can someone suggest an example of a series of functions that converges absolutely but does not converges uniformly?

Answer 1

Let $$f_n(x)=x^n.$$

$\sum f_n(x)$ is absolutly convergent at $[0,1)$ as a geometric one.

it doesn't converge uniformly at $[0,1)$ cause $$\sup_{x\in[0,1)}|\sum_{k=0}^n x^k-\frac{1}{1-x}|=+\infty$$.

Answer 2

Here's one that converges absolutely pointwise everywhere on the compact set $[0,1]:$ $\sum_{n=1}^{\infty}n^2x^n(1-x).$ The convergence isn't uniform on $[0,1],$ because the summands don't converge uniformly to $0.$ In fact the $n$th summand evaluated at $1-1/n$ is $n(1-1/n)^n \to \infty.$