Question

Evaluate: $\underset{n\to \infty }{lim}\sum _{x=1}^{20}{\cos }^{2n}(x-10)=$

• A. $0$
• B. $1$
• C. $20$
• D. $7$

Answer 1

Answer: (B)

$1$

${lim}_{n\to \infty }{\sum }_{x=1}^{20}{\cos }^{2n}(x-10)$

$={lim}_{n\to \infty }{\cos }^2(-9)+{\cos }^2(-8)+-+{\cos }^2\left(0\right)+\cdots$

$sinccos\theta liesbetween-1and1$

$-1⩽\cos \theta ⩽1$

Thus ${lim}_{n\to \infty }{\cos }^{2n}\theta =0$ if $\cos \theta \ne 1$

$\Rightarrow {lim}_{n\to \infty }{\cos }^2(-9)=0$

similariy${lim}_{n\to \infty }{\cos }^{2n}(-8)=0--$

$but{lim}_{n\to \infty }{\cos }^{2n}\left(0\right)={lim}_{n\to \infty }1=1$