Question

obtain the half Range Fourier series of f(x) = l-x in (0,l)​

Answer 1

Answer:

Define f as a periodic function of 1 on R as

f(x)=(x−1)2,0≤x<1

Set x=0, then

f(0)=1=13+∑n=1∞4n2π2

So we have

23=∑n=1∞4n2π2andπ26=∑n=1∞1n2

And set x→1−, then

limx→1−f(x)=0=13+∑n=1∞4n2π2cosnπ

Thus

π212=∑n=1∞1(2n−1)2−∑n=1∞1(2n)2=∑n=1∞1(2n−1)2−14∑n=1∞1n2=∑n=1∞1(2n−1)2−π224

So

∑n=1∞1(2n−1)2=π28