To evaluate $\lim_{n\to\infty} \frac{10^n}{n!} .$

Question

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$

I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to prove it.

Answer 1

$$\frac{10^n}{n!} = \frac{10\cdot 10 \cdot 10 \cdot 10 \cdots 10}{1 \cdot 2 \cdot3 \cdots n} =\frac{10^{11}}{11!}\frac{10}{12}\frac{10}{13}\cdots\frac{10}n < \frac{10^{11}}{11!}\left(\frac{5}{6}\right)^{n-11} \to 0$$ as $n \to \infty.$

Answer 2

The series $\sum_{n=0}^\infty \frac{10^n}{n!}$ is convergent and the sum is $e^{10}$. The convergence can be checked by ratio test $$\lim_{n \to \infty} \frac{10^{n+1}}{(n+1)!}\frac{n!}{10^n}=\lim_{n \to \infty}\frac{10}{n+1}=0<1$$

By the necerssery condition for the convergence of an infinite series the common term must go to 0.

Answer 3

@kmitov gives a very good prove by using ratio test. () Also, you could always remember that factorial grows faster then exponential function. So the limit goes to 0. For a prove:

Set $a_n=10^n/n!$ and take arbitrary $\epsilon>0$, you can always find a $N$ large enough such that for all $n>N$ such that $$ a_n<\epsilon $$ Together with the fact that $a_n\geq 0$ always, so $a_n\to 0$ in the end.

Answer 4

If $n>20$, then $$ 0<a_n=\frac{10^n}{n!}<\frac{10^{20}}{20!} \left(\frac{10}{20}\right)^{n-20}=\frac{20^{20}}{20!}\cdot\frac{1}{2^{n}}\to 0, $$ as $w^n\to$, whenever $\lvert w\rvert<1$.

Answer 5

Another way to do: Let $a_n=\frac{10^n}{n!}$. Clearly $$a_{n+1}=\frac{10}{n+1}a_n$$ and hence $\{a_n\}$ is positive and decreasing when $n\ge 10$. By the bounded Monotone Principle, $\lim_{n\to\infty}a_n=L$ exists. Thus $$ L=\lim_{n\to\infty}a_{n+1}=\lim_{n\to\infty}\frac{10}{n+1}a_n=0\cdot L=0.$$