SEEMOUS 2010 Problem 1

Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by $$f_n(x)=\int^x_0f_{n-1}(t)dt$$ for all integers $n\ge1$ .

a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$ . b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$ .