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Limit of sequence

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the limit of [itex]n^2(e^\frac{1}{n^2} - cos(\frac{1}{n}))[/itex]

    2. Relevant equations



    3. The attempt at a solution

    since cos(1/n) is asymptotic to 1. [itex]n^2(e^\frac{1}{n^2} - cos(\frac{1}{n}))[/itex] ~ [itex]n^2(e^\frac{1}{n^2} - 1)[/itex] ~ [itex]n^2 \frac{1}{n^2})[/itex] = 1
    The right answer is 3/2 though. I don't see what's wrong with my reasoning. Maybe i used asymptotic in an illegitimate way. What's the problem?
     
  2. jcsd
  3. Aug 25, 2011 #2

    CompuChip

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    You have to be a little more careful than that.
    Try switching over to x = 1/n, then it will be the limit for x going to zero.
    If you expand both terms inside the brackets in a series around 0, you can throw away terms of order x4 and you will arrive at the right answer.
     
  4. Aug 25, 2011 #3
    Thanks, that way i solved it.
    I also found what i did wrong with asymptotic. I though that when a sequence is asymptotic with another you could just substitute one with the other. But it's not true. in this case. [itex]cos(1/n) [/itex]~ [itex]1[/itex] but [itex]e^\frac{1}{n^2} - cos(\frac{1}{n})[/itex] ~[itex] \frac{3}{2} e^\frac{1}{n^2} - 1[/itex].

    [itex]e^\frac{1}{n^2} - cos(\frac{1}{n})[/itex] ~[itex]e^\frac{1}{n^2} - 1[/itex] This is not true.
     
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