# Math Help - convolution fourier series question

1. ## convolution fourier series question

P_r is defined as:
$P_r(x)=\frac{1-r^2}{1-2r\cos x +r^2}$
and
$P_r(x)=\frac{1-r^2}{1-2r\cos x +r^2}=\sum_{n=-\infty}^{\infty}r{|n|}e^{inx}$
and
$f(x)=\sum_{-\infty}^{\infty}c_ne^{inx}$
which is continues

i need to prove that:
$f_r(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}p_r(t)dt=\sum_{n=1}^{\infty}c_nr^{|n|}e^ {inx}$

the solution says to use the convolution property
$c_n(f)=c_n$
$c_n(P_r)=r^{|n|}$
$c_n(f_r)=c_n r^{|n|}$

but i cant see how the multiplication of those coefficient gives me the
expression i needed to prove

?

i only got the right side not the left integral

??