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Convergence of a sequence

Source: Romanian District Olympiad 2014, Grade 12, P2

June 15, 2014
functioncalculusderivativeintegrationreal analysisreal analysis unsolved

Problem Statement

Let f:[0,1]→Rf:[0,1]\rightarrow{\mathbb{R}} be a differentiable function, with continuous derivative, and let sn=∑k=1nf(kn) s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) Prove that the sequence (sn+1−sn)n∈N∗(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}} converges to ∫01f(x)dx\int_{0}^{1}f(x)\mathrm{d}x.
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