Page 491, #15,17. Plot the truncated Fourier series for n=2,4,8.
15.
Since 
, the function is odd.
![{\displaystyle a_{0}=(1/\pi )*[-1\int _{-\pi }^{-\pi /2}(x+\pi )dx+\int _{-\pi /2}^{\pi /2}(x)dx+\int _{\pi /2}^{\pi }(\pi -x)dx]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3515ec49fdd8ef95aedb6e74ee844ba7b39b6ce)
After simplification, 
.
Using Equation (5) on page 490, the exapansion becomes;
With 
this becomes;
for n is "odd".
This means when n is even, 
17.
![{\displaystyle a_{0}=1/2[\int _{-1}^{0}(1+x)dx+\int _{0}^{1}(1-x)dx]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/663792d020231d208ed32b73f0c4db7045053632)
![{\displaystyle a_{0}=1/2[1-1/2+1-1/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57762d639f689e1bcd7d58eeb81c3590c3473f73)
![{\displaystyle a_{n}=1/1[\int _{-1}^{0}(1+x)cos(n\pi *x/1)dx+\int _{0}^{1}(1-x)cos(n\pi *x/1)dx]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc54c05805473116b69ac5965dd79d368827f34)
Simplifying results in;
![{\displaystyle a_{n}=2[1/(n^{2}\pi ^{2})_{(}-1)^{n}/(n^{2}\pi ^{2})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82a4babc3b94dfce840755f8704669ff666085ce)
So,
when n is odd.
![{\displaystyle b_{n}=1/1[\int _{-1}^{0}(1+x)sin(n\pi *x/1)dx+\int _{0}^{1}(1-x)sin(n\pi *x/1)dx]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3a3c83592e4975230374d189014a6e9669263f)
Simplifying results in;
Since 
for even n, 
becomes 
for even n.
