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Series and factorial

  1. Mar 29, 2005 #1
    I’ve been playing around with the infinite series:
    [tex] \sum_{k=1}^\infty \frac{k}{(k+1)!} [/tex]

    I haven’t really gotten anywhere with it however I punched it into my calculator and it determined the sum to be 1. And the sum of n terms of the series equals
    [tex]1-\frac{1}{(n+1)(n!)} [/tex]
    Why is this so? Any help is much appreciated.
  2. jcsd
  3. Mar 29, 2005 #2
    Use induction on that last statement. Show its true for n = 1, then assume it's true for n = k, and show it's true for n = k+1
  4. Mar 29, 2005 #3
    I see how I can use induction to find why [tex]1-\frac{1}{(n+1)(n!)} [/tex]
    gives the sum of the series but how would you analytically come up with that expression in the first place. My calculator did it in a second, how did it generate the expression. Is there something I am missing?
  5. Mar 29, 2005 #4


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    Science Advisor
    Homework Helper

    It's a telescoping series, this may help:


    For the infinite series you can also consider:

  6. Mar 29, 2005 #5
    Thanks shmoe, I lost my negative and made the series, dare I say, even more infinite. Mwahahaha...
  7. Oct 31, 2010 #6
    Whay about : \sum_{n=1}^{\infty}\frac{8^{n}}{(n)!} ( I copy like this cause i don´t know how to put the symbol)Does anybody know how to solve this? PLease, help.
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