Are term #$n$ and term #$n+1$ always the highest terms in the Maclaurin series for $e^n$?

Question

Seems to work when I test it empirically (Python code below):

from math import factorial

for n in range(1,10):
    print n,[float(n**k)/factorial(k) for k in range(n+2)]

Is this guaranteed to hold for any positive integer $n$, and if yes, why?

Thank you.

Answer 1

Yes, they are.

For a fixed $n$, consider how we get from term number $k$ to term number $k+1$: $$ \frac{n^k}{k!}\to \frac{n^{k+1}}{(k+1)!} $$ We see that multiply by $\frac n{k+1}$. This is larger than $1$ as long as $k$ is smaller than $n$, so for those $k$ the sequence increases, and for $k$ larger than $n$ what we multiply by is smaller than $1$, so the sequence decreases.

For the transition from term number $n$ to term number $n+1$, we multiply by $\frac{n+1}{n+1} = 1$, so those two terms are equal, and indeed the maximum.