find the fourier series representing f(x)=xsinx

Question

Please help with finding the limits for the series.

Answer 1

f(x) = x.sin(x)

TheFourier series for f(x) = x.sin(x) is given by

s(x) = a0/2 + Sum[n=1 to infinity]{a_n.cos(nx) + b_n.sin(x)}

where

a_n = (1/pi)*int[0 .. 2pi] f(x).cos(nx) dx,  (n >= 0)

b_n = (1/pi)*int[0 .. 2pi] f(x).sin(nx) dx,  (n >= 1)

Evaluating the above coefficients for n = 0 to 3 gives us,

a0 = -2

a1 = -1/2     b1 = pi
a2 = 2/3      b2 = 0
a3 = 1/4      b3 = 0

The fourier series for f(x) = x.sin(x) is

s(x) = -1 - (1/2)cos(x) + pi.sin(x)+ (2/3)cos(2x) + (1/4)cos(3x) + ...