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function sequence, n-repeated integral

Source: SEEMOUS 2010 P1

June 17, 2021
functionConvergenceSequencesintegrationcalculus

Problem Statement

Let f0:[0,1]→Rf_0:[0,1]\to\mathbb R be a continuous function. Define the sequence of functions fn:[0,1]→Rf_n:[0,1]\to\mathbb R by fn(x)=∫0xfn−1(t)dtf_n(x)=\int^x_0f_{n-1}(t)dt for all integers n≥1n\ge1.
a) Prove that the series ∑n=1∞fn(x)\sum_{n=1}^\infty f_n(x) is convergent for every x∈[0,1]x\in[0,1]. b) Find an explicit formula for the sum of the series ∑n=1∞fn(x),x∈[0,1]\sum_{n=1}^\infty f_n(x),x\in[0,1].
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