Competition problem — Algebra — Maths Olympiad Prep

Library / /81 of 264

Algebra Difficulty 5.4 AIME, harder Prove it Romania

For each positive integer $n$ the function $f_n : [0, n] \to \mathbb{R}$ is defined by $f_n(x) = \arctg(\lfloor x \rfloor)$ . Prove that $f_n$ is a Riemann integrable function

and find
$\lim_{n \to \infty} \frac{1}{n} \int_{0}^{n} f_{n}(x) dx.$

Solution

The function $f_n$ is locally constant, hence Riemann integrable.

Next, we have
$\int_{0}^{n} f_{n}(x) dx = \sum_{i=0}^{n-1} \int_{i}^{i+1} f_{n}(i) dx = \sum_{i=0}^{n-1} \arctan i.$
Applying Stolz-Cesàro theorem, we obtain
$\lim_{n \to \infty} \frac{\arctan 1 + \arctan 2 + \dots + \arctan n}{n} = \lim_{n \to \infty} \arctan n = \frac{\pi}{2}.$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.