How can I calculate the Fourier series of $\frac{1-r^2}{1-2r\cos x+r^2}$ when $|r|<1$? I know the answer is $1+\sum_{n=1}^{\infty}2r^n\cos nx$, but is there any way to solve the problem in the realm of real numbers? Thank you for your help.
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1Is it OK to treat the answer as a power series for $r$? Using the fact that the answer is a power series for $r$ converging absolutely for $|r| < 1$, multiplying $1-2rcos(x)+r^2$ to it and then splitting it to 3 power series, replacing $2cos(nx)cos(x)$ with $cos(n+1)x + cos(n-1)x$, it’s not hard to see the answer is correct, although this way is kind of tricky – onRiv Aug 19 '22 at 05:16
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Yeah, that works. Thanks! – carol Aug 19 '22 at 09:24