Summation of $c\sum_{x=0}^{\infty} e^{tx}x$

Question

How can I find the sum of c$\sum_{x=0}^{\infty} e^{tx}x$?

c is a constant in this case.

I am trying to find the moment generating function, but I am stuck on this simplification step.

Answer 1

If $|e^t| <1$ then $$\sum_{x=0}^{\infty} e^{tx} =\sum_{x=0}^{\infty} {(e^{t})}^{x}= \frac{1}{1-e^t}. \tag 1$$

Therefore, $$\frac{d}{dt}\sum_{x=0}^{\infty} e^{tx} = \frac{d}{dt}\left( \frac{1}{1-e^t}\right)=?$$