How do you prove ex * ey = ex + y using series?

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The discussion focuses on proving the identity e^x * e^y = e^(x+y) using series. It suggests that if the series for a_n and b_n converge absolutely, then the series c_n, defined as the convolution of a_n and b_n, also converges absolutely to the product of the sums of the two series. A hint is provided regarding the multiplication of series terms and how it relates to the indices summing to different values. Additionally, a request for the formula for e^x is made to clarify the proof. The conversation emphasizes the connection between series convergence and the multiplication of exponential functions.
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How do you prove ex * ey = ex + y using series?



Prove:

suppose sum an and sum bn converge absolutely then the series cn with cn = sum (from k=0 to n) ak*bn-k converges absolutely to the limit sum an * sum bn

Thank you!

Or if you have seen either of these on here could you redirect me to that page
 
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For the first one, what is the formula for e^x? That should make the proof obvious.

For the second one, I don't know, and dinner's ready, but it probably follows from using the epsilon-delta definition of a limit.
 
Hint: (a0+a1)*(b0+b1)=a0*b0 + (a0*b1+a1*b0) + a1*b1. In the first term the indices sum to 0, in the second to 1 and in the last to 2. Do you see how that is related to what you are asking?
 

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