Summation with exponential functions

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The discussion focuses on proving formulas related to exponential functions and geometric series. The user, a retired individual, seeks assistance with terms involving \((\frac{1}{z})^k\) and \(e^z\). It is noted that \(e^{kz} = (e^z)^k\) and a relationship involving \((-1)^{k-1}\) is also mentioned. The geometric series formula \(\sum_{k=0}^\infty ar^k = \frac{a}{1 - r}\) is suggested as a useful tool for the proof. Overall, the conversation emphasizes the mathematical relationships and series involved in the problem.
Belgium 12
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Dear members,

see attached pdf file.Can you help me to prove this formulas.

Thank you

Belgium 12

This is not homework.I'm 68 and retired.
 

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The terms are a composition of \left(\frac{1}{z}\right)^k and e^z . There should be a nice geometric series formula for this.
 
Yes. e^{kz}= (e^z)^k and (-1)^{k-1}e^{kz}= -((-1)e^z)^k.

So use the fact that the geometric series \sum_{k=0}^\infty ar^k is a/(1- r).
 

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