What property of summation is used in this simplification?

Question

I am looking at a problem that simplifies $\sum_{n=0}^\infty\frac{(t\mu)^n}{n!}$ to $e^{t\mu}$. I can't seem to recall what property this is. Does anyone recognize this?

Answer 1

The Taylor series for $e^x$ is given by

$$e^x = \sum\limits_{k = 0}^{\infty} \frac {x^k}{k!}$$

This is absolutely convergent for all complex $x$, and the simplification follows from setting $x = t\mu$.