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in Mathematics by (60 points)

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Suppose you have the function:
f(x)=excosx
and you need to find the 3rd degree Taylor Series representation. The way I have been taught to do this is to express each separate function as a power series and multiply as necessary for the 3rd degree. For example for
cosx=∑n=0∞(−1)nx2n(2n)!=1−x22!+x44!+⋯ and ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯
multiply the terms on the right of each until you get the 3rd degree.

Logically, I am happy. However, I have not seen a theorem or any rule that says you can just multiply series in this way. Doing it this way, is there a guarantee that I will always get the power series representation of f(x)?

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