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Convergence of Infinite Series

  1. Nov 26, 2007 #1
    Question: Test for convergence:

    [tex]\sum\frac{n!}{10^n}[/tex]

    (the sum is from 1 to infinity)

    I tried using

    [tex]\frac{n^n}{10^n}\geq\frac{n!}{10^n}\geq\frac{n}{10^n}[/tex]

    and showing that either the first one was convergent or the last one was divergent using various tests but didn't get anywhere.

    Any hints?
     
  2. jcsd
  3. Nov 26, 2007 #2
    Try using the Ratio Test. That's what I first try to do anytime I see n!
     
  4. Nov 26, 2007 #3
    Ratio Test:

    [tex]\frac{(n+1)!}{10^{n+1}}\frac{10^n}{n!}=\frac{n+1}{10}[/tex]

    Since that's more than 1 as n goes to infinity it diverges. Am I right?
     
  5. Nov 26, 2007 #4
    I did a bunch of these. I need to get a good grade so I will just post them up here with my answers and if I got one wrong please just let me know to look over it again.

    1. [tex]\sum\frac{1}{ln(n)}[/tex] (sum from 2 to inf.) diverges

    b/c 1/ln(n)>1/n which diverges

    2. [tex]\sum\frac{1}{2n(2n+1)}[/tex] converges

    b/c 1/n^2 converges

    3. [tex]\sum\frac{1}{(n(n+1))^{.5}}[/tex] diverges

    b/c 1/n diverges and then limit comparison test to show this diverges too.

    4. [tex]\sum\frac{1}{2n+1}[/tex] diverges

    b/c 1/2n diverges and then limit comparison to show this diverges too.

    I'd really appreciate it if somebody could check me on these. Thanks.
     
  6. Nov 26, 2007 #5

    Office_Shredder

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    Those all look right to me
     
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