# Infinite Series Expansions

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Is there any way to pre-determine whether a series exists which will satisfy the requirement of fulfilling a given chosen function?

For example, given Sin 3X - Cot 5X / Csc X , does an infinite series exist which satisfies it? How does one know?

Perhaps my question is naively foolish; I studied LaPlace Transforms, Test Ratio Tests, etc., but never truly understood.

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A function can be represented as a power series if and only if it is complex differentiable in an open set. This follows from the general form of Taylor's theorem for complex functions.

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Being real differentiable--even infinitely many times--is not enough, as the function $$e^{\frac{-1}{x^2}}$$ on the real line (equal to 0 at 0) is $$C^\infty$$  yet does not equal its power series expansion since all its derivatives at zero vanish. The reason is that the complexified version of the function is not even continuous at the origin.

Edited by wtf

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