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Does the series with terms n!/e^n converge or diverge

  • Thread starter Ki-nana18
  • Start date
  • #1
91
0

Homework Statement


Does this series converge or diverge? infinity[tex]\Sigma[/tex]n=1 (n!/e^n)


Homework Equations



The ratio test.

The Attempt at a Solution


lim n--> infinity ((e^n)(n+1)!)/(e^(n+1))

I don't know what to do from here...
 

Answers and Replies

  • #2
33,108
4,804


You're missing the n!. Set up the ratio of a_(n+1)/a_n, simplify it, then take the limit.
 
  • #3
91
0


This is what I have:

lim n-->infinity [(n+1)!/(e^(n+1))][(e^n)(n!)]
lim n-->infinity [(n+1)/(e)]

Did I simplify right? And do I just plug in infinity now?
 
  • #4
33,108
4,804


Yes, that's simplified correctly. No, you don't actually plug in infinity, but as n gets larger and larger, what happens to (n + 1)/e?
 
  • #5
91
0


It also gets larger. Which means that the series diverges.
 
  • #6
33,108
4,804


Yes and yes. Using the ratio test, you found that lim a_(n + 1)/a_n is infinity, and so the ratio test tells us this series diverges.
 
  • #7
91
0


Yay! Thank you!
 

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