I'm studying uniform convergence, and am looking for some examples of series $\sum f_n(x)$ that converge absolutely and uniformly in $[a,b]$, but $\sum |f_n(x)|$ does not converge uniformly in the same interval. So far I've found things like $\sum (1-x)x^{n}(-1)^n$ in $[0,1]$.
I don't want to start a big list; I'll accept whichever answer gives a couple of insightful examples.
Thanks!