Series question

  • #1

Homework Statement


[tex]\Sigma[/tex]n!/n^n

index n=1 to infinity


Homework Equations





The Attempt at a Solution


Using the Ratio test (limit as n goes to infinity of a[tex]_{n+1}[/tex]/a[tex]_{n}[/tex])
and found that the series converges.

However, I thought that factorials grew faster than exponential functions. Therefore, it would diverge, right?

Could someone explain why? Did I just do something wrong?
 

Answers and Replies

  • #2
ptr
28
0
Exponents of the form n^n grows much faster than factorials of the form (n!) because the factorial is a multiplication of n terms, the majority of which are less than n, and the power is a multiplication of n terms, all of which are equal to n.
 
  • #3
35,616
7,492
Factorials don't grow faster than exponentials of the sort you're working with. Just think about it: n! = 1 * 2 * 3 * ... * n. You have n factors, of which the largest is n.
n^n = n * n * n * ... * n. Here you have n factors, all of which are n. Clearly this exponental is larger than the factorial above.
 
  • #4
Avodyne
Science Advisor
1,396
90
Exponentials with a fixed base, like e^n or 2^n, grow more slowly than n!.
 

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