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Can you help with this problem?

Thank you so much!

I want to multiply this two series:

Sin Power Series

Geometric Power Series

This is the result:

enter image description here

I want to do exactly like this example that is at the end, but I do not understand how to combine them. I do not understand also how in the example they found (k-1)/2 and all the other details in the result.

As they are power series of analytic functions, the result is also analytic in the intersection of the intervals (-1,1)

enter image description here

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    If you want go straight for the solution, substitute $2j+1$ by $(2j+1)!$ in the double summation at the end. The inner sum comes from the grouping of coefficients of terms with the same degree (from the product of the two series) and, because exponents in the series for arctan have the same progression ($2k+1$) of exponents as for sin, the affirmation follows. – 2017-02-27
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    @RafaBudría Thank you for your help! I had guessed that before to post my question... However, I want to know why or what are the rules when you are combining the series, if they exist... – 2017-02-28
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    There are rules for that. – 2017-02-28

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