may i know how to prove the theorem that factorials beat exponentials?

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- May 9th 2010, 11:57 AMalexandrabel90factorials
may i know how to prove the theorem that factorials beat exponentials?

- May 9th 2010, 12:18 PMDrexel28
- May 9th 2010, 01:22 PMalexandrabel90
sorry, i dont really get what you are trying to hint to me. how do i make use of this formula to show that

eg the fraction of an exponential over factorial will tend to 0? - May 9th 2010, 01:34 PMDrexel28
- May 9th 2010, 01:40 PMBruno J.
Here's a much simpler way. We know converges for every real number , which immediately implies .

- May 9th 2010, 01:49 PMDrexel28
- May 9th 2010, 02:00 PMalexandrabel90
isit because n(n+0.5) can be taken as n^n thus the fraction becomes (a.e/n)^n where (a.e/n) is less than 1?

- May 9th 2010, 02:01 PMalexandrabel90
in my course that im learning, i have yet to learn the formula for factorials. so assuming that i dont know that formula, is there another method to prove it?

thanks - May 9th 2010, 02:32 PMdwsmith
- May 9th 2010, 03:04 PMRenji Rodrigo
Proposition[ratio teste for sequences]

Iff (x_n) is a sequence with x_n >0 forall n in N

and

then .

Take a>0

then

so

then

- May 9th 2010, 03:06 PMBruno J.
- May 9th 2010, 03:07 PMBruno J.
- May 9th 2010, 03:21 PMalexandrabel90
by the way, is there a way to use sandwich theorem to prove this?

- May 9th 2010, 03:45 PMRenji Rodrigo
Given a>0 exists with ,

so for we have

taking the product in from to we have

taking the product with on the both sides off the inequality

the limit on the right goes to infinity so the limit on the left too

, so