Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 28


$$\sum_{i=1}^{n+1}\sin \left(\frac{\pi}{i}\right)$$

Work Step by Step

Given $$\sin (\pi)+\sin \left(\frac{\pi}{2}\right)+\sin \left(\frac{\pi}{3}\right)+\ldots+\sin \left(\frac{\pi}{n+1}\right)$$ Since the first term is $ \sin (\pi)$ and the last is $\sin \left(\frac{\pi}{n+1}\right)$, then $$\sin (\pi)+\sin \left(\frac{\pi}{2}\right)+\sin \left(\frac{\pi}{3}\right)+\ldots+\sin \left(\frac{\pi}{n+1}\right)=\sum_{i=1}^{n+1}\sin \left(\frac{\pi}{i}\right)$$
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