Finding the asymptotics of $\sum_{k=1}^n a^k k!$? Note that $a > 0$.

Question

There's no way to use integration method in this case. I also tried to use Stolz–Cesàro theorem, but couldn't find right $y_n$. What method should I use?

Answer 1

I've got it! Ok, so here we go:

First, factor $a^n n!$ out of the sum:

$$\sum_{k=1}^n a^k k! = a^n n! ( 1 + \frac{1}{an} + \frac{1}{a²n(n-1)} + \cdots ) $$

It is clear that the inner bracket tends to 1 as n $ \to \infty $, so the asymptotic behaviour is indeed given by $a^n n! $