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Homework Help: Limit that equals e^2

  1. May 3, 2012 #1
    1. The problem statement, all variables and given/known data
    the problem asks to find the limit of the sequence an = (1+(2/n))^n

    2. Relevant equations

    3. The attempt at a solution
    i saw a solution where they said:

    lim{n→∞} [1+(2/n)]^n
    = lim{n→∞} [1+(2/n)]^[(n/2)2]
    = [lim{m→∞} [1+(1/m)]^m]^2, where m = n/2
    = e^2

    am i supposed to just know that [1+(1/m)]^m = e ? or am i supposed to show it for this problem
  2. jcsd
  3. May 3, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The limit as m goes to infinity is e. Either you're supposed to know this or you're supposed to prove it... what is the definition of e that you are using?
  4. May 3, 2012 #3


    Staff: Mentor

    Your book probably has an example where they let y = (1 + 1/m)^m, then take ln of both sides.

    After that they take the limit and use L'Hopital's Rule. There's a little more to it than I've said, but that should give you something to look for.
  5. May 3, 2012 #4
    i dont know what the definition of e is that im using ...

    i just dont know in my hw if i could just get to that point and then say,oh yea [1+(1/m)]^m = e and that be it. or if i actually have to show it. i guess it doesnt hurt to show it. ill try it out
  6. May 4, 2012 #5


    Staff: Mentor

    You don't need the definition of e. Use what I said in post #3.
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