Multiplying McLaurin Functions & Power Series

jaejoon89
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How do you multiply McLaurin functions by "regular" power series?
For example:

If y' = sum (1, inf.) n*a_n x^n-1 and cos(x) = sum(0, inf.) (-1^n x^2n) / (2n)!, how do you find the product?

If y = sum(0, inf.) a_n x^n and sin(x) = sum(0, inf.) (-1^n x^(2n+1)) / (2n+1)!, how do you find the product of sin(x), y, and x?
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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