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calculate limit using integral

Source: Romanian District Olympiad 2000, Grade XII, Problem 4

September 25, 2018
functionslimitsreal analysiscontestsromaniacalculusintegration

Problem Statement

Let f:[0,1]⟶R+∗ f:[0,1]\longrightarrow\mathbb{R}_+^* be a Riemann-integrable function. Calculate lim⁡n→∞(−n+∑i=1ne1n⋅f(in)). \lim_{n\to\infty}\left(-n+\sum_{i=1}^ne^{\frac{1}{n}\cdot f\left(\frac{i}{n}\right)}\right) .
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