may i know how to prove the theorem that factorials beat exponentials?
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Originally Posted by alexandrabel90 may i know how to prove the theorem that factorials beat exponentials?
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?
Originally Posted by alexandrabel90 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? Why?
Here's a much simpler way. We know converges for every real number , which immediately implies .
Originally Posted by Bruno J. Here's a much simpler way. We know converges for every real number , which immediately implies . How do you know the OP has seen series? Really all you tacitly assumed was that if then it's dominated by a convergent geometric sequence.
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?
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
Originally Posted by Drexel28 How do you know the OP has seen series? What does OP stand for?
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
Originally Posted by Drexel28 How do you know the OP has seen series? Really all you tacitly assumed was that if then it's dominated by a convergent geometric sequence. How do you know the OP has seen Stirling's formula? You're right about all that I've assumed, which is comparatively quite little.
Originally Posted by dwsmith What does OP stand for? "Original Poster"
by the way, is there a way to use sandwich theorem to prove this?
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
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