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, ![{\displaystyle a_{0}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f3589226b1f07bd27b7c82d8f470a4685fffe2)
.
Using Equation (5) on page 490, the exapansion becomes;
![{\displaystyle f(x)=8k/\pi ^{2}[sin(\pi *x/L)-sin(3\pi *x/L)/9...)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee6df1febd26a27d67e60bb464aa6fbf8454a94)
With
this becomes;
for n is "odd".
This means when n is even, ![{\displaystyle f(x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1)
![](//upload.wikimedia.org/wikiversity/en/2/27/R63a.jpg)
17.
![{\displaystyle f(x)=1-|x|,-1\leq x\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3bd7723ea233d07072066bba050abdd0eb9726)
![{\displaystyle f(x)=\left\{{\begin{matrix}1+x-\pi ,-1<x<0\\1-x,0<x<1\\\end{matrix}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db1bab6f543a90c4fd1b800fcc0a1391bcd45d31)
![{\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_{0}=1/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82733c82a9a3f4fd483529c221322e82c679eb61)
![{\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;
![{\displaystyle b_{n}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc141d40ca4a13ec7beafd6d264d5a435792a58)
![{\displaystyle f(x)=1/2+\sum _{n=0}^{\infty }4/(n^{2}\pi ^{2})*cos(n\pi *x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/352fe9408d8d2a57b7b7b5a62039e90f176fe464)
Since
for even n,
becomes
for even n.