Convolution fourier series question

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

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

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

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

??
 

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