Why does $\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$? Here is a standard identity:
$$\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$$
Why does it hold true?

The Taylor series for $e^x$ is
$$
\sum_{k = 0}^{\infty} \frac{x^k}{k!}.
$$
Letting $x = a$, we obtain
$$
e^a = \sum_{k = 0}^{\infty} \frac{a^k}{k!}
$$
as desired.
