Infinite Sum

Calculus Level 2

True or False?

$\quad$ The series $\displaystyle \sum_{n=1}^\infty \left ( \cos\left( \dfrac \pi n\right) - \cos\left( \dfrac \pi{n+2}\right) \right)$ converges.

True False

1 solution

Rishabh Jain
Jun 27, 2016

Relevant wiki: Telescoping Series - Sum

Given is a telescopic series which comes out to be: $\cos (\pi)+\cos\left(\dfrac{\pi}2\right)-\lim_{n\to\infty}\left(\cos\left(\dfrac{\pi}{n+1}\right)+\cos\left(\dfrac{\pi}{n+2}\right)\right)$

$=-1+0-(1+1)=\boxed{-3}$ Hence series converges to $-3$ .

Thanks Rishabh

Jose Sacramento - 4 years, 11 months ago

Welcome .............

Rishabh Jain - 4 years, 11 months ago

There is a typo...replace n+1 by n in the limit n tends to infinity

Ravi Dwivedi - 4 years, 11 months ago

Doent matter...Although its correct since I calculated partial sum till n terms.

Rishabh Jain - 4 years, 11 months ago

Infinite Sum

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