Question

can you help me understand the process for this question?

Answer 1

We are given the following series:

$\sum ^9_{k\mathop=0}\sin (\pi k)$

this means:

$\sin (\pi\times0)+\sin (\pi\times1)+\sin (\pi\times2)+\sin (\pi\times3)+\cdots+\sin (\pi\times9)$

since the since of any multiple of pi is zero, this means that all these terms are zero, therefore, their sum is zero, this means:

$\sin (\pi k)=0\text{ for every k}$

therefore:

$\sum ^9_{k\mathop=0}\sin (\pi k)=0$