(adsbygoogle = window.adsbygoogle || []).push({}); Question about the "start" of a cosine fourier series

Hey. I was just looking through Paul's Online Notes http://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx to teach myself Fourier Series and I had a question about the a[itex]_{0}[/itex] term of the cosine series.

In the online lesson, it says assume an even function has the series f(x) = [itex]\Sigma[/itex]a[itex]_{n}[/itex]cos(n[itex]\pi[/itex]x/L) where -L≤x≤L. The series starts at 0, and the way Paul gave a prove of it was to multiply the series by cos(m[itex]\pi[/itex]x/L) and then integrated and used the fact that cos(m[itex]\pi[/itex]x/L) and cos(n[itex]\pi[/itex]x/L) were orthogonal if m!=n.

So that for example, for the fourier series of x[itex]^{2}[/itex], he got a[itex]_{0}[/itex] = L[itex]^{2}[/itex]/3, where -L≤x≤L.

However, my question is, why do we need to start at n = 0? The proof using orthogonality would work just as well if n were to start at 1 or 100, and the formula for the coefficients would remain the same. In addition, I'm not sure convergence explains it, since the beginning terms of a infinite series have no effect on the convergence of an infinite series. So I was hoping someone could clear this up for me.

Thanks in advance!

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# Question about the start of a cosine fourier series

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