series convergence and divergence

**Legendsn3verdie** · November 30th 2008, 02:12 PM

hey first of all i d like to start this problem off with saying i have no idea what the ! means so i cant even start this problem.. even if the ! wasnt there i d assume this was a geometric series since its in that section of the book. Other then that i know if it converges i need to find the sum. any help appreciated.

**skeeter** · November 30th 2008, 02:23 PM

n! ... is read "n" factorial.

e.g.

$5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$

$n! = n(n-1)(n-2)(n-3)...(3)(2)(1)$

familiar with the ratio test for convergence/divergence ?

**Mathstud28** · November 30th 2008, 03:10 PM

Originally Posted by Legendsn3verdie

hey first of all i d like to start this problem off with saying i have no idea what the ! means so i cant even start this problem.. even if the ! wasnt there i d assume this was a geometric series since its in that section of the book. Other then that i know if it converges i need to find the sum. any help appreciated.

$\frac{n^n}{n!}=\frac{\overbrace{n\cdot{n}\cdot{n}\ cdots}^{n\text{ number of times}}}{\underbrace{n\cdot(n-1)\cdots}_{n\text{ number of times}}}=\frac{n}{n}\cdot\frac{n}{n-1}\cdots\geqslant{1\cdot{1}\cdots=1}$

So we can see that

$\sum_{n=1}^{\infty}1\to\infty\leqslant\sum_{n=1}^{ \infty}\frac{n^n}{n!}$

**galactus** · November 30th 2008, 03:19 PM

$\sum_{n=1}^{\infty}\frac{n^{n}}{n!}$

The ratio test may be a decent choice here.

${\rho}=\lim_{n\to +\infty}\frac{u_{n+1}}{u_{n}}=\lim_{n\to +\infty}\frac{(n+1)^{n+1}}{(n+1)!}\cdot\frac{n!}{n ^{n}}$

$=\lim_{n\to +\infty}\frac{(n+1)^{n}}{n^{n}}$

$=\lim_{n\to +\infty}\left(1+\frac{1}{n}\right)^{n}=e$

Since e>1, then it diverges.

If you do not recognize it, that last limit is a very famous one. Therefore, one does not have to bother proving it every time.

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