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Limit Question using the definition of e

  1. Nov 8, 2014 #1

    RJLiberator

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

    2*Lim (as k approaches infinity) of (| (k/(k+1))^k |)

    The answer to this limit is 2/e

    I know there is a definition of e used, but I am unclear what to do/how to do it. If someone has a link I can look at or could point me in the right direction I would be thankful.
     
  2. jcsd
  3. Nov 8, 2014 #2

    Dick

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    The definition of e that you want is that it's the limit k->infinity (1+1/k)^k. That's pretty closely related to your limit.
     
  4. Nov 8, 2014 #3

    RJLiberator

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    Ugh. I'm thinking of all the possible ways to transform this limit.

    If I take a K out of every part it becomes:
    (1/k)^k/(1+1/k)^k and the denominator can be replaced with e, but the numerator then goes to 0.

    Am I on the right track? The limit (k-->infinity) of (1/k)^k = 0 so that can't be right.

    Hmmm...

    Somehow symbolab changes the equation into (k/(k+k))^k to make this work, but I have no idea how you can change 1 to k.
     
  5. Nov 8, 2014 #4

    Dick

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    You are doing bad algebra. (1/k)/(1+1/k) isn't equal k/(k+1).
     
  6. Nov 8, 2014 #5

    RJLiberator

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    Ah. I see.
    (1/k)/(1+1/k) = 1/(k+1)

    This is interesting. I feel this is getting extremely close to the goal.
    However, it puzzles me. All we did was take a k out of every element of the equation to make the limit easier.
    We want to correlate this to the definition of e.

    If I make the definition:
    1/e^k = 1/(1+1/k)^k

    I seem to get closer to what my answer states, but not quite there.
     
  7. Nov 8, 2014 #6

    Dick

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    Just fix the algebra. If you have k/(k+1) and you take a k out of numerator and denominator what do you get?
     
  8. Nov 8, 2014 #7

    RJLiberator

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    Ugh... well, that was easier then I made it seem.
    My algebra was off from the beginning.

    Clearly, taking a K out of the numerator and denominator makes it the simple equation of
    [1/(1+1/k)]^k which fits the definition of e flawlessly
    1/e^k = [1/(1+1/k)]^k

    Marvelous. You've been a great help here.
     
  9. Nov 8, 2014 #8

    Dick

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    I really hope you meant to say 1/e=limit k->infinity 1/(1+1/k)^k. That's kind of different from what you posted.
     
  10. Nov 8, 2014 #9

    RJLiberator

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    ;)Precisely.
    Thank you for pointing that out - helps me understand the definition of e, better.
     
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