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Homework Help: Limit of a Sequence

  1. Apr 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Show convergence or divergence, find the limit of the sequence.

    2. Relevant equations

    [tex]

    a_{n}=\sqrt[n]{4^{n}n}

    [/tex]

    3. The attempt at a solution

    See the attached picture. Sorry, it was taking way longer to figure out how to input the TeX than it was to just snap and crop a picture of my work.

    My question is: where did I lose the coefficient 4? The correct answer is supposed to be convergent, a_n --> infinity = 4
     

    Attached Files:

  2. jcsd
  3. Apr 16, 2010 #2
    In the step
    [tex]\lim_{x\rightarrow\infty} y = 4\lim_{x\rightarrow\infty}x^{\frac{1}{x}} \Leftrightarrow \lim_{x\rightarrow\infty}\ln y = 4\lim_{x\rightarrow\infty}\ln x^{\frac{1}{x}}[/tex]
    you forgot to take the logarithm of the 4 on the right hand side.
     
  4. Apr 16, 2010 #3
    True, but how does that affect the outcome? The limit(x-->infinity) of 1/x = 0 thus negating anything else that multiplies against it, which still leads me back to e^0 = 1.

    NvM, I see that I need to put the 4 back inside the limit then with log properties it becomes ln4+ln(1/x)... e^ln4 = 4... therefore a_n-->4. It's not always obvious to me when I should factor constants out or just leave them be.

    Thanks for pointing that out, NeoDevin!
     
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