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Homework Help: Am i doing this right?/convergence of sequences

  1. Oct 9, 2011 #1
    1. The problem statement, all variables and given/known data
    [tex]a_n=(\frac{1}{n})^{\frac{1}{\ln{n}}}[/tex]


    2. Relevant equations



    3. The attempt at a solution

    [tex]\lim_{x\rightarrow \infty}(\frac{1}{n})^{\frac{1}{\ln{n}}}[/tex]
    [tex]y=(\frac{1}{n})^{\frac{1}{\ln{n}}}[/tex]
    [tex]\ln{y}=\frac{\ln{\frac{1}{n}}}{\ln{n}}[/tex]
    [tex]\ln{y}=\frac{\ln{1}-\ln{n}}{\ln{n}}[/tex]
    [tex]\ln{y}=\frac{\ln{1}}{\ln{n}}-\frac{\ln{n}}{\ln{n}}[/tex]
    [tex]\ln{y}=0-1[/tex]
    [tex]e^{\ln{y}}=e^{-1}[/tex]
    [tex]\lim_{x\rightarrow \infty}(\frac{1}{n})^{\frac{1}{\ln{n}}}=e^{-1}[/tex]

    the back of my book says the answer is indeed [itex]e^{-1}[/itex] but im not sure if this is the way to go to for the solution or whether im supposed to rewrite it as something similar to [itex]\lim_{x\rightarrow \infty}(1+\frac{x}{n})^{n}[/itex]
     
  2. jcsd
  3. Oct 9, 2011 #2

    LCKurtz

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    Homework Helper
    Gold Member

    You steps are fine. That problem gives a constant sequence, just written in a non-obvious way.
     
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