Infinite series...Convergence/Divergence

Mar 2010
6
0
Limit...Infinity...factorial

Hello,
Is the limit n!/n^4 as n goes to infinity , equals to infinity?
Thanks
 
Last edited:
Oct 2009
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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

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

Clearly tonio has merely misread the problem. The answer to your question is yes.
 
Oct 2009
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So \(\displaystyle \lim\text{ }\frac{n^4}{n!}=0\overset{?}{\implies}\lim\text{ }\frac{n!}{n^4}\)?

Clearly tonio has merely misread the problem. The answer to your question is yes.

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

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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

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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

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\(\displaystyle \frac{1}{\frac{-1}{n}}\not\to\infty\)

(I'm in a terse mood, haha)

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

Tonio
 

Drexel28

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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.