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Homework Help: Convergent series

  1. Jan 23, 2008 #1
    Hi, could you please check if my solution is correct?

    1. The problem statement, all variables and given/known data

    Test the following series for convergence:

    [tex]\sum_{n=1}^{\infty}\frac{1!+2!+...+n!}{(\left 2n \right)!}[/tex]

    3. The attempt at a solution

    I can use a slightly altered series

    [tex]\sum_{n=1}^{\infty}\frac{nn!}{(\left 2n \right)!} [/tex]

    whose every term is >= than the corresponding term in the original series.. and thus if this altered series converges, then the original one should so as well...

    Then, if I use the limit ratio test for the second series:

    [tex]\lim_{n \rightarrow \infty}\frac{(\left n+1\right)(\left n+1 \right)!}{(\left 2n+2 \right)!}\frac{(\left 2n\right)!}{n(\left n \right)!} = 0 [/tex]

    This means that the altered series is convergent, and thus the original series is also convergent.

    Is this reasoning correct? Thanks in advance!
    Last edited: Jan 23, 2008
  2. jcsd
  3. Jan 23, 2008 #2
    Yes it is.
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