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Well-known trigonometric series bound

Source: 2017 South Korea USCM P7

August 15, 2020

seriesreal analysistrigonometric seriesinequalitiescollege conteststrigonometry

Problem Statement

Prove the following inequality holds if

\{a_n\}

is a deceasing sequence of positive reals, and

0<\theta<\frac{\pi}{2}

\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}