TITLE: Partial sums of exponential series
QUESTION [22 upvotes]: What is known about $f(k)=\sum_{n=0}^{k-1} \frac{k^n}{n!}$ for large $k$?
Obviously it is is a partial sum of the series for $e^k$ -- but this partial sum doesn't reach close to $e^k$ itself because we're cutting off the series right at the largest terms. In the full series, the $(k+i-1)$th term is always at least as large as the $(k-i)$th term for $1\le i\le k$, so $f(k)< e^k/2$. Can we estimate more precisely how much smaller than $e^k$ the function is?
It would look very nice and pleasing if, say, $f(k)\sim e^{k-1}$ for large $k$, but I have no real evidence for that hypothesis.
(Inspired by this question and my answer thereto).

REPLY [16 votes]: $e^{-k} f(k) = P(X_k < k)$ where $X_k$ is a Poisson random variable with parameter $k$.  Since the sum of $k$ independent Poisson random variables with parameter $1$ is Poisson with parameter $k$, the Central Limit Theorem says the probability distribution of $Z_k = (X_k - k)/\sqrt{k}$ approaches a standard normal distribution.  Thus $P(X_k < k) = P(Z_k < 0) \to 1/2$ as $k \to \infty$.