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Asymptotic behaviour of 1^1*2^2*...*n^n

Source: IMC 2024, Problem 2

August 7, 2024
logarithmsreal analysiscalculus

Problem Statement

For n=1,2,…n=1,2,\dots let Sn=log⁡(11⋅22⋅…⋅nnn2)−log⁡(n),S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}), where log⁡\log denotes the natural logarithm. Find lim⁡n→∞Sn\lim_{n \to \infty} S_n.
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