Question

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

• A. $0$
• B. $1$
• C. $20$
• D. $14$

Answer 1

The correct option is B $1$

B)

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

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

since caso lies between $-1$ and 1

$-1⩽\cos \theta ⩽1$

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

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

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

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