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limit integral with distance to nearest integer

Source: Putnam 1983 B5

September 17, 2021
calculusintegrationlimits

Problem Statement

Let ∥u∥\lVert u\rVert denote the distance from the real number uu to the nearest integer. For positive integers nn, let an=1n∫1n∥nx∥dx.a_n=\frac1n\int^n_1\left\lVert\frac nx\right\rVert dx.Determine lim⁡n→∞an\lim_{n\to\infty}a_n.
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