International Mathematics Competition
for University Students
2024

IMC2024: Day 1, Problem 2

Problem 2. For \(\displaystyle n=1,2,\ldots\) let

\(\displaystyle S_n = \log\left( \sqrt[n^2]{1^1\cdot 2^2\cdot\ldots\cdot n^n}\right) - \log (\sqrt{n}), \)

where \(\displaystyle \log\) denotes the natural logarithm. Find \(\displaystyle \mathop{\lim}\limits_{n\to\infty} S_n\).

Sergey Chernov, Belarusian State University, Minsk

Hint: \(\displaystyle S_n\) is (close to) a Riemann sum of a certain integral.

  Solution