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sequences of f:[0,1]->R, sum_k(f(k/n))

Source: SEEMOUS 2011 P4

June 18, 2021
Sequencesintegrationcalculus

Problem Statement

Let f:[0,1]→Rf:[0,1]\to\mathbb R be a twice continuously differentiable increasing function. Define the sequences given by Ln=1n∑k=0n−1f(kn)L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right) and Un=1n∑k=0nf(kn)U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right), n≥1n\ge1. 1. The interval [Ln,Un][L_n,U_n] is divided into three equal segments. Prove that, for large enough nn, the number I=∫01f(x)dxI=\int^1_0f(x)\text dx belongs to the middle one of these three segments.
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