Series $\sum \limits_{k=1}^{\infty}f_{n}(x)$ converges uniformly, but $\sum \limits_{k=1}^{\infty}|f_{n}(x)|$ converges only pointwise

Question

Is there a sequence $f_{n}:\mathbb{R} \to \mathbb{R}$ such that the series $\sum \limits_{k=1}^{\infty}f_{n}(x)$ converges uniformly to $f(x)$, but $\sum \limits_{k=1}^{\infty}|f_{n}(x)|$ converges only pointwise, for all $x\in \mathbb{R}$?

I tried limiting the domain to $(0,1)$ and using Dirichlet Eta function, but that approach failed. I know that the sequence $f_{n}$ cannot be strictly positive and can't be equibounded, otherwise $\sum \limits_{k=1}^{\infty}|f_{n}(x)|$ would converge uniformly by Weierstrass M-test.

Answer 1

Try $$f_n(x) = \frac{(-1)^n}{n} , 0 < xn<1$$