MathDB

2

Part of 2024 IMC

Problems(1)

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

Source: IMC 2024, Problem 2

8/7/2024
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.
logarithmsreal analysiscalculus