# Infinite series...Convergence/Divergence

#### alireza6485

Limit...Infinity...factorial

Hello,
Is the limit n!/n^4 as n goes to infinity , equals to infinity?
Thanks

Last edited:

#### tonio

Hello,
Is the limit n!/n^4 as n goes to infinity , equals to infinity?
Thanks

Applying for example the quotient test (D'alembert's test), we see that the series $$\displaystyle \sum^\infty_{n=1}\frac{n^4}{n!}$$ converges, so...

Tonio

#### Drexel28

MHF Hall of Honor
Applying for example the quotient test (D'alembert's test), we see that the series $$\displaystyle \sum^\infty_{n=1}\frac{n^4}{n!}$$ converges, so...

Tonio
So $$\displaystyle \lim\text{ }\frac{n^4}{n!}=0\overset{?}{\implies}\lim\text{ }\frac{n!}{n^4}$$?

#### tonio

So $$\displaystyle \lim\text{ }\frac{n^4}{n!}=0\overset{?}{\implies}\lim\text{ }\frac{n!}{n^4}$$?

Hmmm...I'm afraid clearly Drexel has misread Tonio's solution: since the positive infinite series $$\displaystyle \sum^\infty_{n=1}a_n$$ converges then $$\displaystyle a_n\xrightarrow [n\to\infty]{}0\Longrightarrow \frac{1}{a_n}\xrightarrow [n\to\infty]{}\infty$$ .

Tonio

#### Drexel28

MHF Hall of Honor
Hmmm...I'm afraid clearly Drexel has misread Tonio's solution: since the positive infinite series $$\displaystyle \sum^\infty_{n=1}a_n$$ converges then $$\displaystyle a_n\xrightarrow [n\to\infty]{}0\Longrightarrow \frac{1}{a_n}\xrightarrow [n\to\infty]{}\infty$$ .

Tonio
$$\displaystyle \frac{-1}{n}\to 0$$

#### matheagle

MHF Hall of Honor
write out a few terms....

For n greater than some number, like 10 or 100...

$$\displaystyle {n!\over n^4}>\left({n(n-1)(n-2)(n-3)\over n^4}\right)(n-4)>c(n-4)\to\infty$$

where c is some positive number.
If you say for n>10, then let c=1/8 easily works.

#### Drexel28

MHF Hall of Honor
Yes, indeed...so?

Tonio
$$\displaystyle \frac{1}{\frac{-1}{n}}\not\to\infty$$

(I'm in a terse mood, haha)

#### tonio

$$\displaystyle \frac{1}{\frac{-1}{n}}\not\to\infty$$

(I'm in a terse mood, haha)

Yes indeed, again...so? The given series' general term was positive, as I pointed out.

Tonio

#### Drexel28

MHF Hall of Honor
Yes indeed, again...so? The given series' general term was positive, as I pointed out.

Tonio
Just saying more needed to be said haha, I didn't mean this to turn into a ten post thing.