Are Abel's and Dirichlet tests necessary for uniform convergence?

Question

Can a series of functions $$\sum_{n=1}^{\infty} {f_n(x)}$$ that doesn't meet the conditions of Abel's (or Dirichlet) test be uniformly convergent?

Answer 1

Both of these criteria require you to write $f_n(x)$ in the form $a_n(x)b_n(x)$ where one of these new sequences of functions is monotonically decreasing. So to prevent these tests from being applicable, you just need to deny the possibility of finding a useful monotonically decreasing sequence of functions. For example if $$ f_n(x):=\begin{cases} \frac1n&\text{if $n\le x<n+1$}\\ 0&\text{otherwise} \end{cases} $$ then $\sum_nf_n(x)$ converges uniformly. However, the $f_n$'s are nonzero on non-overlapping intervals, so if you write $f_n=a_nb_n$ with $b_n$ monotonically decreasing, then the $a_n$ you wind up with isn't materially different from $f_n$: proving uniform convergence (or uniform boundedness) for $a_n$ is no easier than for $f_n$. In other words, for this counterexample the Abel and Dirichlet criteria don't offer any help.