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Convergence Test Problem

  1. Sep 27, 2013 #1
    1. The problem statement, all variables and given/known data
    determine whether the Ʃ n4 / en2 is convergent or divergent?


    2. Relevant equations



    3. The attempt at a solution
    Using Root test:
    lim of n4/n / en as n approaches infinity
    But lim of n4/n as n approaches infinity = ∞0
    So: Let N = lim of n4/n as n approaches infinity
    and: ln N = lim of 4ln(n)/n as n approaches infinity = ∞/∞
    By Lhopitals rule: ln N = lim 4/n as n approaches infinity = 0
    thus ln N = 0 ; 1 = N
    Therefore: lim of n4/n / en as n approaches infinity = 1/∞ = 0
    thus CONVERGE?

    Is this solution Ok?
     
  2. jcsd
  3. Sep 27, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    The notation could be improved (might be the conversion to text here), but the method is fine.
    Be careful with the domain of n when you check the limit of n^(4/n): you want the limit for natural numbers, but then you treat n as a real number. This is possible (a limit for real numbers for n->infinity is also a limit for natural numbers), but you have to consider it.
     
  4. Sep 27, 2013 #3
    I think it's correct. For practice, try using the ratio test using inequalities.
     
  5. Sep 27, 2013 #4
    Are there another way on how to test the convergence of this series?
     
  6. Sep 27, 2013 #5
    Yes. Take the ratio test. Then use the definition of Euler's constant

    [itex]e=\lim_{n\rightarrow \infty}\left(n+\frac{1}{n}\right)^n[/itex]

    to find where the limit lies
     
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