# Math Help - find the limit of the sequence

1. ## find the limit of the sequence

find the limit of the sequence: (n!) ^ (1/n) as n -> infinity

i don't even know where to start on this one. i think if i get a nod in the right direction i'd be able to get it, so can anyone here give me that nod?

2. It could help to consider $\frac{1}{n}ln(n!)\;=\;\frac{1}{n}(ln(n)+ln(n-1)+...+ln(2))$

3. $\sqrt[n]{n!}\approx{\frac{n}{e}}$ as $n\rightarrow{+\infty}$

Simply: $e^n=1+n+\frac{1}{2}n^2+...+\frac{n^n}{n!}+...$

Thus: $e^n\geq{\frac{n^n}{n!}}$ and $n!\geq{\frac{n^n}{e^n}}$ (this is enough for what you want to know, but do go on)

And: $\int_1^{n+1}{\ln(x)dx}\geq{\sum_{k=1}^{n}{\ln(i)}} =\ln(n)!$

Then $(n+1)\ln(n+1)-n\geq{\ln(n)!}$ and $\frac{(n+1)^{(n+1)}}{e^n}\geq{n!}$

So we have $\frac{(n+1)^{n+1}}{e^n}\geq{n!}\geq{\frac{n^n}{e^n }}$

Therefore: $\frac{(n+1)^{1+\frac{1}{n}}}{e}\geq{\sqrt[n]{n!}}\geq{\frac{n}{e}}$

Now if you divide by $\frac{n}{e}$ and take the limit you get that, by the squeeze theorem $\sqrt[n]{n!}\approx{\frac{n}{e}}$

4. Use the following trick.

Let $a_n$ be a seqeuence of non-zero real numbers. If $\left| \frac{a_{n+1}}{a_n}\right|$ converges then $\left| a_n \right|^{1/n}$ converges to the same thing.