Question

${lim}_{n\to \infty }\frac{\cos x+\cos 3x+\cos 5x+\cos (2n-1)x}{n}$ where $x\ne k\pi ,kϵ$

Answer 1

$\underset{n\to \infty }{lim}\left[\frac{\cos \left(x\right)+\cos \left(3x\right)+\cos \left(5x\right)+\cos (2n-1)x}{n}\right]$

Apply the sequence theorem

$-1\le \cos (2n-1)\le 1$

$\underset{n\to \infty }{lim}\left(\frac{\cos \left(x\right)+\cos \left(3x\right)+\cos \left(5x\right)+(-1)x}{n}\right)\le \underset{n\to \infty }{lim}\left(\frac{\cos \left(x\right)+\cos \left(3x\right)+\cos \left(5x\right)+\cos (2n-1)x}{n}\right)$

$\le \underset{n\to \infty }{lim}\left[\frac{\cos \left(x\right)+\cos \left(3x\right)+\cos \left(5x\right)+1x}{n}\right]=0$

Apply Infinity property : $\underset{x\to \infty }{lim}(\frac{c}{x^a}=0)$

$\underset{n\to \infty }{lim}\left[\frac{\cos \left(x\right)+\cos \left(3x\right)+\cos \left(5x\right)+1.x}{n}\right]=0$

$\therefore$ By the sequence theorem

$\underset{n\to \infty }{lim}\left[\frac{\cos \left(x\right)+\cos \left(3x\right)+\cos \left(5x\right)+\cos (2n-1)x}{n}\right]$

$=0$