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Math Help - Converges or Diverges?

  1. #1
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    Converges or Diverges?

    I need some help determining whether or not the equation in the attachment diverges or converges. I'm trying to teach myself the integral test, and cannot seem to get this one to work. Any help would be much appreciated.
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  2. #2
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    Quote Originally Posted by rman144 View Post
    I need some help determining whether or not the equation in the attachment diverges or converges. I'm trying to teach myself the integral test, and cannot seem to get this one to work. Any help would be much appreciated.
    So in other words you are asking for

    \int_1^{\infty}\bigg[\arctan\left(\frac{a}{x+\frac{1}{2}}\right)-\frac{a}{x}\bigg]?
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  3. #3
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    No, I'm saying that "x" can be any value greater than one and less than positive infinity
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    So in other words you are asking for

    \int_1^{\infty}\bigg[\arctan\left(\frac{a}{x+\frac{1}{2}}\right)-\frac{a}{x}\bigg]?
    Quote Originally Posted by rman144 View Post
    No, I'm saying that "x" can be any value greater than one and less than positive infinity
    Yeah, Did you notice that I changed the variables, it won't change the result.

    You want to show that

    \int_1^{\infty}\bigg[\arctan\left(\frac{x}{n+\frac{1}{2}}\right)-\frac{x}{n}\bigg]~dn<\infty\quad\forall{x}>1
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    Sorry, I'm kind of new to this stuff. Yeah, you are correct. Still, how can I prove it diverges/converges?
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by rman144 View Post
    Sorry, I'm kind of new to this stuff. Yeah, you are correct. Still, how can I prove it diverges/converges?
    Lets look at the integral

    \int\bigg[\arctan\left(\frac{a}{x+\frac{1}{2}}\right)-\frac{a}{x}\bigg]~dx

    =\int\bigg[\frac{\pi}{2}-\arctan\left(\frac{x+\frac{1}{2}}{a}\right)-\frac{a}{x}\bigg]

    Now we can see that the first and last terms are

    \frac{\pi{x}}{2} and a\ln|x| respectively.

    Now let us consider

    \int\arctan\left(\frac{x+b}{a}\right)~dx

    If we use integration by parts once we get

    \int\arctan\left(\frac{x+b}{a}\right)~dx

    =x\arctan\left(\frac{x+b}{a}\right)-a\int\frac{x}{(x+b)^2+a^2}

    Now if we let z=x+b\Rightarrow{x=z-b}

    and

    dx=dz

    So this would gives us

    a\int\frac{z-b}{z^2+a^2}~dz

    =a\int\frac{z}{z^2+a^2}~dz-ab\int\frac{dz}{z^2+a^2}

    Now by letting z^2+a^2=u and z=a\tan(\theta) respectively we come up with

    \frac{a}{2}\ln(z^2+a^2)-b\arctan\left(\frac{z}{a}\right)

    Now making the appropriate backsubs we get

    \frac{a}{2}\ln\left((x+b)^2+a^2\right)-b\arctan\left(\frac{x+b}{a}\right)

    Putting this all back together we get

    \int\arctan\left(\frac{x+b}{a}\right)

    =x\arctan\left(\frac{x+b}{a}\right)-\frac{a}{2}\ln\left((x+b)^2+a^2\right)+b\arctan\le  ft(\frac{x+b}{a}\right)

    =\left(x+b\right)\arctan\left(\frac{x+b}{a}\right)-\frac{a}{2}\ln\left((x+b)^2+a^2\right)+C

    So putting this completely back together and remembering that b=\frac{1}{2} we get

    \int\bigg[\arctan\left(\frac{a}{x+\frac{1}{2}}\right)-\frac{a}{x}\bigg]~dx

    =\frac{\pi{x}}{2}-\left(x+\frac{1}{2}\right)\arctan\left(\frac{x+\fr  ac{1}{2}}{a}\right)+\frac{a}{2}\ln\left(\left(x+\f  rac{1}{2}\right)^2+a^2\right)-a\ln|x|



    Can you go from there?
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    Yeah, thanks a lot. This site saves lives
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  8. #8
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by rman144 View Post
    Yeah, thanks a lot. This site saves lives
    It's not the site, its the people that volunteer their time here.

    EDIT: You also might want to consider combinging the last two natural logs so that it is more obvious as the integrals behavior as x tends toward infinity.
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    Thanks, already done.
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  10. #10
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    I know that this is beating a dead horse, but I seem to still have a problem. The integral test keeps telling me the series converges, but the graph of the equation is basically telling me that it diverges. Am I just messing myself up on this one, or did I make a mistake with the integral test?
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  11. #11
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by rman144 View Post
    I know that this is beating a dead horse, but I seem to still have a problem. The integral test keeps telling me the series converges, but the graph of the equation is basically telling me that it diverges. Am I just messing myself up on this one, or did I make a mistake with the integral test?
    The integral converges.
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