I am trying to prove that if f(x) and g(x) both have period P, then f(x)*g(x) also has period P.(adsbygoogle = window.adsbygoogle || []).push({});

What I tried to do was let f(x) and g(x) be represented as Fourier series

[tex]f(x)=a_{0,1}+\sum_{n=1}^{\infty}\left[a_{n,1}cos(\frac{2{\pi}n}{P}x)+b_{n,1}sin(\frac{2{\pi}n}{P}x)\right][/tex]

[tex]g(x)=a_{0,2}+\sum_{n=1}^{\infty}\left[a_{n,2}cos(\frac{2{\pi}n}{P}x)+b_{n,2}sin(\frac{2{\pi}n}{P}x)\right][/tex]

I then tried to multiply the right side of these 2 equations and then manipulate it to look like a Fourier series with constants [tex]a_{0,3},a_{n,3},b_{n_3}[/tex]. Multiplying it out became problematic when the last term became a product of two series:

[tex]\left(\sum_{n=1}^{\infty}\left[a_{n,1}cos(\frac{2{\pi}n}{P}x)+b_{n,1}sin(\frac{2{\pi}n}{P}x)\right]\right)*\left(\sum_{n=1}^{\infty}\left[a_{n,2}cos(\frac{2{\pi}n}{P}x)+b_{n,2}sin(\frac{2{\pi}n}{P}x)\right]\right)[/tex]

Is there any easy way to simplify this...If not, is there a better strategy for approaching this problem? Thanks.

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# Periodic function proof

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