Uniform convergence of $∑ f_n$ along with pointwise convergence of $ ∑ | f_n(x)| $ implies uniform convergence of $∑ | f_n(x)|$? If true, prove it; if not true, give a counterexample.
I don't think this is right but I have trouble thinking about a counterexample. Could you please think of any? Thanks a lot!
Sketch: For $x\in [0,1],$ the power series
$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}.$$
converges uniformly to $\ln(1+x)$ on $[0,1].$ That requires some proof (Dirichlet's theorem for uniform convergence will do it), but after all this is just a sketch.
Now let's restrict to $[0,1).$ Then this power series is absolutely convergent on $[0,1).$ But the absolute value series, namely
$$\sum_{n=1}^{\infty}\frac{x^n}{n},$$
does not converge uniformly on $[0,1).$