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If I have some Fourier Series representation of a function with $x$ period of $2L$

$$G(x,y) = \sum_{n = 1}^{\infty} \left[a_n \sin\left(\frac{n \pi x}{L}\right) + b_n \cos\left(\frac{n \pi x}{L}\right)\right]f_n(y)$$

Where each of the $f_n(y) \to 0 $ as $y \to \infty$ do we necessarily have that

$$\lim_{y \to \infty} G(x,y) = 0$$

I do not know the relevant analysis -- maybe we are not always allowed to take termwise limits of an infinite sum? Is there a general result that applies here? How might I go about calculating limits of sums like this, is there a particular method that is usually applied?

Thank You

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up vote 1 down vote accepted

Well, guess in your case you are correct. You can see for example "Elementary Properties of Limits of Functions" (Ch.6, p.156) in S. Krantz Real Analysis and Foundations.

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