Question

The value of $\underset{n\to \infty }{\text{lim}}\sum _{x=1}^{20}{\cos }^{2n}(x-10)$ is equal to

• A. $0$
• B. $1$
• C. $19$
• D. $20$

Answer 1

The correct option is B $1$

$\underset{n\to \infty }{\text{lim}}{\cos }^{2n}x=\{\begin{array}{ccc}1,& x=r\pi ,& r\in I\\ 0,& x\ne r\pi & r\in I\end{array}$

Here, for $x=10,\underset{n\to \infty }{\text{lim}}{\cos }^{2n}(x-10)=1$ and in all other cases, it is zero.

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