# Limit of Multivariable Fourier Series

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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