Question

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

Answer 1

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

Answer 2

hey sandra
you dont have a better reason?

Answer 3

i dont buy this argument

Answer 4

how about consider x! / x^x

Answer 5

for your father

Answer 6

what you wrote isnt obvious , sorry

Answer 7

we can make x! continuous using gamma function

Answer 8

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

Answer 9

well you didnt show it converges

Answer 10

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

Answer 11

hehe

Answer 12

Did you guys try the ratio test?

Answer 13

i for one did not :p

Answer 14

ratio test is for series, this is a sequence

Answer 15

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

Answer 16

how about a sandwich theorem

Answer 17

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

Answer 18

so by induction we have

Answer 19

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

Answer 20

err

Answer 21

it is strictly less than 1

Answer 22

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

Answer 23

ok so divide out the first n

Answer 24

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

Answer 25

the root solution is way over my head

Answer 26

sandra i made you a hero

Answer 27

omg lol! yay =)

Answer 28

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

Answer 29

How do you give points to people?

Answer 30

you "become their fan"

Answer 31

I think they should add per answer type ratings

Answer 32

at least in addition

Answer 33

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

Answer 34

I can't find "become their fan"

Answer 35

this is a tough question

Answer 36

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

Answer 37

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

Answer 38

\[x + x = 2xx\]

Answer 39

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

Answer 40

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