Solution
= Solution
An Erlang$(n,1)$ variable is a sum of $n$ independent rate-one exponential variables, each with mean and variance one. The central <limit> theorem therefore gives
$$
\frac{X-n}{\sqrt n}\Rightarrow N(0,1).
$$
Consequently, for every fixed $q\geq0$,
$$
\boxed{\int_0^{n+q\sqrt n}\frac{x^{n-1}e^{-x}}{(n-1)!}dx
=\mathbb P\left(\frac{X-n}{\sqrt n}\leq q\right)
\longrightarrow\Phi(q).}
$$