Chapter F, Problem 24E

To determine

To find: The value of the sum ${\displaystyle \sum _{k=0}^8\cos k\pi }$.

Answer to Problem 24E

The value of the sum ${\displaystyle \sum _{k=0}^8\cos k\pi }$ is $1$.

Explanation of Solution

Definition used:

If $a_m,a_{m+1},\mathrm{...},a_n$ are real numbers and m and n are integers such that $m\le n$, then ${\displaystyle \sum _{i=m}^n{a}_i=a_m+}{a}_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_n$.

Calculation:

By the above definition, the sum ${\displaystyle \sum _{k=0}^8\cos k\pi }$ expressed as follows.

$\begin{array}{c}{\displaystyle \sum _{k=0}^8\cos k\pi }=\left\{\begin{array}{c}\cos \left(0\pi \right)+\cos \left(1\pi \right)+\cos \left(2\pi \right)+\cos \left(3\pi \right)+\cos \left(4\pi \right)+\cos \left(5\pi \right)+\cos \left(6\pi +\right)+\\ \cos \left(7\pi \right)+\cos \left(8\pi \right)\end{array}\right\}\\ =1-1+1-1+1-1+1-1+1\\ =1\end{array}$

Thus value of the sum ${\displaystyle \sum _{k=0}^8\cos k\pi }$ is $1$.