Chapter 8.3, Problem 89E

Fourier Series The following sum is a finite Fourier series.

$\begin{array}{l}f(x)={\displaystyle \underset{i=1}{\overset{N}{\sum }}{a}_i\sin ix}\\ =a_1\sin x+a_2\sin 2x+a_3\sin 3x+\cdots +a_N\sin Nx\end{array}$

(a) Use Exercise 88 to show that the nth coefficient $a_n$ is

given by $a_n=\frac{1}{\pi }{\displaystyle {\int }_{-\pi }^{\pi }f(x)\sin nxdx}$

(b) Let $f(x)=x$. Find $a_1,a_2$, and $a_3$