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Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Consider the sequence $(8^n/n!)n\in \mathbb{N}$. I am trying to show that it is bounded above (it is bounded below by 0) and applying the Squeeze's theorem to show that the limit is 0.

I thought about $2^n/n! < 2(2/n) < 2$

BUt the presence of the upperbound of 2 bothers me.

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marked as duplicate by Pedro Tamaroff, M Turgeon, Hawk, William, J. M. Sep 27 '12 at 10:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Your logic is good. For a different approach, notice that boundedness of a sequence is implied by convergence. – Karolis Juodelė Sep 25 '12 at 20:09
Compare the sequence $\frac{8^n}{n!}$ to $\left(\frac{8^8}{8!}\cdot\frac{8^{n-8}}{9^{n-8}}\right)$ for $n$ bigger than 8. – Arthur Sep 25 '12 at 20:09
up vote 1 down vote accepted

$$0 \leq \frac{2^n}{n!} = \frac{2}{1} \frac{2}{2} \frac{2}{3} \cdots \frac{2}{n} = 2 \cdot \frac{2}{3} \cdots \frac{2}{n} \leq 2 \cdot \left(\frac{2}{3} \right)^{n-2}$$

and since $\frac{2}{3} < 1$, we know $\left(\frac{2}{3} \right)^{n-2} \to 0$ as $n \to \infty$.

For any number replacing 2, say $k$, eventually $n$ is bigger than $k$, so this same argument applies.

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Let $a_n = \frac{8^n}{n!}$. Then for $n \ge 16$, $a_{n+1} < a_n/2$. That's all you need to show that the limit is 0.

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What about a proof using infinite series? Let $\,a>0\,$ and put

$$a_n:=\frac{a^n}{n!}\Longrightarrow \frac{a_{n+1}}{a_n}=\frac{a^{n+1}}{(n+1)!}\frac{n!}{a^n}=\frac{a}{n+1}\xrightarrow [n\to\infty]{} 0$$

So the ratio (D'Alembert's) test tells us the infinite positive series $\,\displaystyle{\sum_{n=1}^\infty a_n}\,$ converges and thus $\,\displaystyle{\lim_{n\to\infty} a_n=0}$ , and this is true for any $\,a>0\,$

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Let $x_n=\frac{a^n}{n!}$ with $a>0$. then all $x_n$ are positive and we have $\frac{x_{n+1}}{x_n}=\frac{a}{n+1}<1$ for $n>a-1$, hence $x_{\lfloor a\rfloor}$ is an upper bound and from then on the sequence decreases monotonuously.

But you can get more: $\frac{x_{n+1}}{x_n}=\frac{a}{n+1}<\frac 12$ for $n>2a-1$, hence $x_n< \frac C{2^n}$ for almost all $n$ for suitable $C$ (choose $C$ such the inequality holds for $n=\lfloor 2a\rfloor$).

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