Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to show that e converges ?

  1. Aug 5, 2010 #1
    Hi All,

    How do we go about showing the euler's number e converges ?
    Recall that

    e =(1+1/n)^n as n ->infinity

    Some place prove this by showing the sequence is bounded above by 3 and is monotonic increasing, thus a limit exist.

    But I forgot how exactly the proof looks like.


    Thanks
    J
     
  2. jcsd
  3. Aug 5, 2010 #2
    Here's one way.

    Write down (1+1/n)^n using the binomial formula.

    You'll see that it equals to

    1 + 1 + (1/2!)(1-1/n) + (1/3!) (1-1/n) (1-2/n) + ... + (1/(n-1)!) (1-1/n) (1-2/n) .. (1-(n-1)/n)

    From here it's not hard to see that it's monotonic increasing, and it's bounded above by 1 + 1 + 1/2! + 1/3! + ... 1/n!.

    In turn, it's not hard to see that the series [itex]\sum 1/n![/itex] converges, because [itex]1/n! < 2^{-n}[/itex] for all n>3.
     
  4. Aug 5, 2010 #3

    lurflurf

    User Avatar
    Homework Helper

    Use the fact that
    geometric mean<=arithmetic mean
    consider n+1 numbers where 1 is one of the numbers and the other n are 1+1/n
     
  5. Aug 5, 2010 #4
    That only proves that it's monotonic.
     
  6. Aug 5, 2010 #5
    Thanks for the neat proof.

    one comment:

    since we know that [itex]\sum 1/n![/itex] converges, we know e converges because
    e = 1 + [itex]\sum 1/n![/itex] where n->infinity
     
    Last edited: Aug 5, 2010
  7. Aug 5, 2010 #6
    but first you'd have to prove that the original series converges and that its limit is equal to 1 + [itex]\sum_{n=1}^{\infty} 1/n![/itex] .
     
  8. Aug 5, 2010 #7

    lurflurf

    User Avatar
    Homework Helper

    2<(1+1/n)^n<(1+1/n)^(n+1)<4

    By Monotone convergence theorem bounded monotonic series converge.
     
    Last edited: Aug 5, 2010
  9. Aug 5, 2010 #8
    I don't see it.
     
  10. Aug 6, 2010 #9

    lurflurf

    User Avatar
    Homework Helper

    ^
    (1+1/n)^(n+1) is decreasing thus
    (1+1/n)^(n+1)<(1+1/1)^(1+1)=4
     
  11. Aug 6, 2010 #10

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You repeated use of "e is convergant" is confusing. Numbers do not converge. It makes sense to ask how to show that the limit [itex]\lim_{n\to\infty} (1+ 1/n)^n[/itex] converges or how to show that [itex]\sum_{n=0}^\infty 1/n![/itex] converges.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook