For the past couple of days, I've been trying to come up with a example for a problem in which $\sum_{k=1}^{\infty} |f_{k}(x)|$ does not converge uniformly but $\sum_{k=1}^{\infty} |f_{k}(x)|$ converges pointwise and $\sum_{k=1}^{\infty} f_{k}(x)$ converges uniformly for all x $\in$ A.
Is my search in vain or is there such a function?
My thought is the series I'm looking for will be of the form $$\sum_{k=1}^{\infty} (-1)^{k}x^{k}$$ since $x^{k}$ does not converge uniformly on $[0,1]$. I'm not sure if it satisfy the conditions.