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Math Help - Integral Test

  1. #1
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    Integral Test

    I have to use the integral test to figure out if the summation of 1/(n2+4) converges or diverges.

    the first thing I did was set it to a function (1/(x2+4)) then I set up an integral from 1 to infinity for that function. Then took the limit as b>infinity from 1 to b for f(x).

    I'm stuck here. The solution manual says that I should get the following:

    lim as b>inf [(1/2)Tan-1(x/2)] from b to 1.

    Can someone explain this to me please? I can figure out the steps after but I don't understand this step.
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  2. #2
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    Re: Integral Test

    Hey bakinbacon.

    You need to consider the integral which should be [(1/2)arctan(x/2)] {x=1,x=infinity} and show that the answer is finite and if so then the series converges by the integral test.

    If it doesn't converge then the series will diverge.
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  3. #3
    MHF Contributor

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    Re: Integral Test

    Which "step" are you talking about?
    Are you okay with [itex]\inti\frac{dx}{a^2+ x^2}= arctan(ax)+ C[/tex]?

    Do you know that tan(x) goes to infinity as x goes to \pi/2? From that, arctan(x) goes to \pi/2 as x goes to infinity.
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