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I'm not sure where to begin on showing that a Fourier series of a periodic function that is neither odd or even can be represented by the sum of the cosine fourier series and sine fourier series.
I know that an even function of period $2L$ is a Fourier cosine series and simiarly with an odd function and sine series. And that a real valued function can be shown to be the sum of an odd function and an even function. Is this little description enough to show it? Suggestions would be great.
Yes, that's enough unless your goal is to show that both sines and cosines will be present (with some nonzero coefficients). For that you can observe that both even and odd parts will be nonzero functions. Alternatively, you can argue that a series of cosines represents an even functions, while a series of sines represents an odd functions; since yours is neither, it must have a series with both sines and cosines.
All of this feels as an unsatisfactory answer, because it is not clear what is a meant by a function here (continuous, an element of $L^2$, etc), and what is the expected level of rigour.
