Absolute and unconditional convergence in $C[0,1]$

Question

I'm trying to come up with example of series from $C[0,1]$ that converges unconditionally, but not absolutely. I know for sure that it exists, but I can't find an example. Firstly I came up with idea of $f_n(x)=(-1)^nx^n$ but it does not converge in $x=1$, so I kinda stuck with this problem. Can anybody help?

Answer 1

Normally we talk of unconditional convergence for a series, not a sequence.

Try this example.

$f_n \ge 0$ has support $[\frac{1}{n+1},\frac{1}{n}]$ and maximum $\frac{1}{n}$. Then the series $\sum f_n$ converges unconditionally to the pointwise limit, but $\|f_n\| = \frac{1}{n}$, so $\sum \|f_n\|$ diverges.