# 2019-06 Simple but not too simple integration

Compute the following integral  $\int_{0}^{\pi/2} \log{ (2 \cos{x} )} dx$.

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# Solution: 2019-05 Convergence with primes

Let $$p_n$$ be the $$n$$-th prime number, $$p_1 = 2, p_2 = 3, p_3 = 5, \dots$$. Prove that the following series converges:
$\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.$

The best solution was submitted by 김기현 (수리과학과 대학원생). Congratulations!

Here is his solution of problem 2019-05.

Other solutions were submitted by 강한필 (전산학부 2016학번), 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기수 (수리과학과 2018학번), 김기현 (수리과학과 대학원생), 김태균 (수리과학과 2016학번), 박항 (전산학부 2013학번), 신원석 (서울대학교 컴퓨터공학부), 이본우 (수리과학과 2017학번, +3), 이정환 (수리과학과 2015학번, +3), 조재형 (수리과학과 2016학번, +3), 채지석 (수리과학과 2016학번), 최백규 (생명과학과 2016학번, +3), 김민서 (2019학번, +2), 윤창기 (서울대학교 화학과).

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# 2019-05 Convergence with primes

Let $$p_n$$ be the $$n$$-th prime number, $$p_1 = 2, p_2 = 3, p_3 = 5, \dots$$. Prove that the following series converges:
$\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.$

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