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HELP! Stuck on an Infinite Series. Thanks

  1. Nov 22, 2011 #1
    I have the infinite series...

    [itex]\sum n^n/n![/itex]

    somehow, I need to use the Ratio Test... and shoe that the resulting limit
    is equal to e. I can't figure out what I am doing wrong algebraically.

    Right now I have simplified this limit to lim((n+1)^n/n^n).

    Please Help...


    Thanks.
     
    Last edited by a moderator: Nov 22, 2011
  2. jcsd
  3. Nov 22, 2011 #2
    THE SERIES IS n^n/n!
     
  4. Nov 22, 2011 #3

    Mark44

    Staff: Mentor

    This is the same as (1 + 1/n)n.

    To evaluate a limit like this, let y = (1 + 1/n)n. Now take the ln of both sides, and then take the limit as n -> infinity. Your textbook should have an example of this type of limit.
     
  5. Nov 22, 2011 #4
    I know how to evaluate this type of limit, my problem is algebraically getting to the point of writing lim((n+1)^n/n^n)=lim(1+1/n)^n. How do I go from my ratio test to seeing the limit as equal to the latter limit?


    Thanks
     
  6. Nov 22, 2011 #5

    Mark44

    Staff: Mentor

    It's fairly basic algebra.
    [tex]\frac{(n + 1)^n}{n^n} = \left(\frac{n+1}{n}\right)^n[/tex]
     
  7. Nov 22, 2011 #6
    Wow, third shift is really getting to me. Thanks Alot.
     
  8. Nov 22, 2011 #7
    You should all ready know this property of exponents:

    [itex]\frac{a^{n}}{b^{n}}[/itex]=([itex]\frac{a}{b}[/itex])[itex]^{n}[/itex]

    Bleh...too late. I fail at tex. lol
     
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