Summation with exponential functions

[Belgium 12]

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.

 

[joeblow]

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.

 

[HallsofIvy]

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 $a/(1- r)$.