I am currently trying to evaluate the following:
$$\lim_{t\rightarrow \infty}\sum_{n=1}^t \left(n-1\right) \dfrac{p^n}{n!}$$
I'm not sure if this is a Taylor series type of equation but it sure looks like one and thus hoping to solve this with Taylor series.
Just pull it apart: \begin{align} \sum\limits_{n=1}^\infty (n-1)\frac{p^n}{n!} &= \sum\limits_{n=1}^\infty n\frac{p^n}{n!} - \sum\limits_{n=1}^\infty \frac{p^n}{n!} \\ &= \left(p\sum\limits_{n=1}^\infty \frac{p^{n-1}}{(n-1)!}\right) - \left(\sum\limits_{n=0}^\infty \frac{p^n}{n!}-1\right)\\ &= p e^p-e^p+1 \\ &= (p-1)e^p+1 \end{align} Here we have used the fact that $e^p = \sum\limits_{n=0}^\infty \frac{p^n}{n!} = \sum\limits_{n=1}^\infty\frac{p^{n-1}}{(n-1)!}$.