1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    For that one I think you need to use the power series expansion of e^x.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Showing That The Infinite Series 1/n! is less than 2
Loading...