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Using the integral test to test for divergence/convergence

  1. Oct 13, 2011 #1
    1. The problem statement, all variables and given/known data
    [tex]\sum_{n=1}^{\infty}\frac{8\arctan{n}}{1+n^2}[/tex]


    2. Relevant equations



    3. The attempt at a solution
    so im comparing it to the integral [itex]\int_{1}^{\infty}\frac{8\arctan{x}}{1+x^2}[/itex]
    but at first i need to show that the function im integrating is continuous, positive and decreasing. I know its continuous and positive from 1 to infinity but i need to show that it is decreasing
    so i found the derivative of the function and got [itex]\frac{8-16x\arctan{x}}{(1+x^2)^2}[/itex] but i got stuck trying to find the critical points
    specifically i forgot how to solve equations like [itex]8-16x\arctan{x}=0[/itex], mainly just because of that extra x in front of the arctan
    i know this boils down to more of a precalc problem but i posted this on the calc homework help because i thought maybe some of my earlier steps were wrong
     
  2. jcsd
  3. Oct 13, 2011 #2

    Mark44

    Staff: Mentor

    I don't see anything wrong with your work.

    The equation you're trying to solve can be simplified to 2x arctan(x) = 1. I don't know of any ways to get analytic solutions to this equation, but approximation techniques yield solutions at [itex]x \approx \pm 0.765379[/itex], according to WolframAlpha. From the graph here, you can say that 1 - 2xarctan(x) < 0 for all x > 1, hence the derivative you've found is negative for x > 1.
     
  4. Oct 13, 2011 #3
    yeah i tried wolfram alpha, but it didnt show any steps on how to solve that equation, was looking for an analytic solution but this will have to do
    thanks mark44
     
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