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Homework Help: Showing That The Infinite Series 1/n! is less than 2

  1. Apr 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the series:
    [itex]\sum\frac{1}{n!}[/itex], where n begins at one and grows infinitely larger (Sorry, I'm still a bit new to the equation editor on here :) )
    1) Use the ratio test to prove that this series is convergent.

    2) Use the comparison test to show that S < 2

    3) Write down the exact value of S.

    2. The attempt at a solution

    The first part of this problem was rather simple.

    However, parts 2 and 3 have me completely stumped. I have tried comparing [itex]\frac{1}{n!}[/itex] to [itex]\frac{1}{n^2}[/itex], but when n = 4, [itex]\frac{1}{n!}[/itex] becomes smaller than [itex]\frac{1}{n^2}[/itex]. Which leads me to believe that this would be true for any series of the form [itex]\frac{1}{n^p}[/itex].

    I have also considered using a geometric series, but, again, I can't think of any that would remain less than [itex]\frac{1}{n!}[/itex]...

    So, what exactly do I compare it too? You don't have to outright give me the answer, but a nudge in the right direction would be nice. And I figure that once I get part 2, part 3 SHOULD fall into place.

    Thanks guys!
  2. jcsd
  3. Apr 21, 2012 #2
    well it is just e-1
  4. Apr 21, 2012 #3
    I am aware that the answer is e-1, but I need to know how to get that. :)
  5. Apr 21, 2012 #4


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    For the comparison part here's a hint: 1/(1*2*3)<1/(1*2*2).
  6. Apr 21, 2012 #5
    Oh, wow...I was totally making this way more difficult than it needed to be. Thanks!
  7. Apr 21, 2012 #6
    So I managed to prove that the sum is less than 2. Now how do I go about finding the exact value of the sum?
  8. Apr 21, 2012 #7


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    For that one I think you need to use the power series expansion of e^x.
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