Given $f_1(x)$ continuous function in $[0,1]$ and differentiable in $(0,1)$. The series {$f_n(x)$} is defined by $f_{n+1}(x)=\int_{0}^{x}f_n(t)dt$ for $x\in [0,1]$.
I need to prove that $\sum_{1}^{\infty}f_n(x)$ converges and is differentiable in $(0,1)$.
I tried to bound $f_n(x)$ by some other series and to use Weierstrass M-Test to prove uniform convergence which will lead to regular one, but I didn't manage to find a series.
I'd really love your help with this one.
Thanks a lot!