does this sequence converge or diverge?
(n!)/(n^n) and the reason

- anonymous

does this sequence converge or diverge?
(n!)/(n^n) and the reason

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- sandra

this converges to zero since eventually n! expands into n*....n-(n-1) [i.e. 1], while the last multiple of n^n is still n. i.e. once n is big enough the last divisor is obviously bigger enough to bring the number to zero

- anonymous

hey sandra
you dont have a better reason?

- anonymous

i dont buy this argument

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## More answers

- anonymous

how about consider x! / x^x

- anonymous

for your father

- anonymous

what you wrote isnt obvious , sorry

- anonymous

we can make x! continuous using gamma function

- sandra

hmmm, not sure about that =) can you explain?

- anonymous

well you didnt show it converges

- sandra

yeah I know, I gave the best reasoning I could heh

- anonymous

hehe

- anonymous

Did you guys try the ratio test?

- sandra

i for one did not :p

- anonymous

ratio test is for series, this is a sequence

- anonymous

or maybe im wrong, i havent see ratio test for sequences

- anonymous

how about a sandwich theorem

- anonymous

n*n >= n * n-1
n*n*n > = n * n-1 n - 2

- anonymous

so by induction we have

- anonymous

n * n-1 * n-2 / n*n*n < = 1

- anonymous

err

- anonymous

it is strictly less than 1

- anonymous

n* n > (n-1)(n-2) for n >= 1

- anonymous

ok so divide out the first n

- sandra

ok, spolier alert... this is a bit beyond my current understanding, but I think the by induction track sounded good... http://www.physicsforums.com/showthread.php?t=195508

- sandra

the root solution is way over my head

- anonymous

sandra i made you a hero

- sandra

omg lol! yay =)

- sandra

I'm already your fan, so not much I can do :p

- anonymous

How do you give points to people?

- sandra

you "become their fan"

- sandra

I think they should add per answer type ratings

- sandra

at least in addition

- sandra

something like "best answer", or "thanks" or something. one more thing to collect :p

- anonymous

I can't find "become their fan"

- anonymous

this is a tough question

- anonymous

see , the series 1/n diverges, but the sequence 1/n converges to zero when you take the limit

- sandra

hmmm jkwon, should be right by the person's name , e.g. right above my reply HERE

- anonymous

\[x + x = 2xx\]

- anonymous

we know n! > n-1 * n-2

- anonymous

i think we can use the fact that its less than 1 and decreasing

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