International Mathematics Competition
for University Students
2020

IMC2020: Day 2, Problem 8

Problem 8. Compute

\(\displaystyle \lim_{n\to \infty} \frac1{\log \log n}\sum_{k=1}^n (-1)^k\binom{n}{k} \log k. \)

(Here \(\displaystyle \log\) denotes the natural logarithm.)

Fedor Petrov, St. Petersburg State University

Hint: Write \(\displaystyle \log k\) as the integral of some function.

  Solution