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Math Help - complicated p-series

  1. #1
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    complicated p-series

    find the values of p for which the series is convergent:

    \sum_{n=3}^{\infty}\frac{1}{n ln n [ln(ln n)]^p}

    please help!
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  2. #2
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    integral test ...

    \int_3^{\infty} \frac{1}{x\ln{x}[\ln(\ln{x})]^p} \, dx

    u = \ln{x}

    du = \frac{1}{x} \, dx

    \int_{\ln{3}}^{\infty} \frac{1}{u(\ln{u})^p} \, du

    consider three cases ... p < 1, p = 1, and p > 1
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    i'm kind of confused because i thought that the integral test was supposed to test for convergence or divergence..but don't you already know it will be convergent somewhere, because it's a p-series? i thought that you couldn't use the evaluation of the integral to tell you where the series actually converges to.. so, should i just evaluate the limit of the improper integral (once i've integrated) and then.. how do the p values i'm supposed to consider come into play?
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  4. #4
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    Quote Originally Posted by buttonbear View Post
    i'm kind of confused because i thought that the integral test was supposed to test for convergence or divergence..but don't you already know it will be convergent somewhere, because it's a p-series? i thought that you couldn't use the evaluation of the integral to tell you where the series actually converges to.. so, should i just evaluate the limit of the improper integral (once i've integrated) and then.. how do the p values i'm supposed to consider come into play?
    the "p" in this problem is not the same as the "p" in the series \sum{\frac{1}{n^p}} ... it's just a variable for the exponent.
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  5. #5
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    so, how can you integrate 1/u(ln u)^p with the two variables? i'm just kind of lost..
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  6. #6
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    Quote Originally Posted by buttonbear View Post
    so, how can you integrate 1/u(ln u)^p with the two variables? i'm just kind of lost..
    \int \frac{1}{u (\ln{u})^p} \, du

    remember substitution?

    t = \ln{u}

    dt = \frac{1}{u} \, du


    \int \frac{1}{t^p} \, dt<br />
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  7. #7
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    honestly, i can't claim i understand any of these 'tools' enough to be able to wield them for this kind of complicated stuff.. thanks for your help
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  8. #8
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    if p=1 then i know it just comes out to be ln t...but if it's >1 or < 1 then it's the power rule... so how do you test if it's convergent? i know it's not right to just come on here and ask for answers, but i'm kind of struggling, do you think you could walk me through it?
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  9. #9
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    \int \frac{1}{t^p} \, dt converges for p > 1, diverges for p \leq 1
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  10. #10
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    thank you so much for your help

    this seems like an awful lot of work if this function/sequence behaves exactly the same way as the p-series
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