Example of series of functions that converges uniformly but whose series of uniform norms does not converge

Question

Analysis two is a very heavy exam, and many people try it thousands of times. Thus, I still have friends to help about that, and today I have been asked, "What do I do when investigating the convergence of a series of functions?". So I started by saying that pointwise convergence is just a bunch of numerical serieses so it's Analysis one, then I mentioned Weierstrass's criterion. If that fails, I am pretty much at a loss. So I was wondering: are there known examples of serieses of functions which converge uniformly on some interval in the real line, but whose uniform norms on that interval have a diverging series?

Answer 1

The alternating harmonic series viewed as a series of constant functions converges uniformly but the series of norms does not converge.