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Math Help - Partial Sums

  1. #1
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    Partial Sums

    http://math.rutgers.edu/~greenfie/we...fstuff/w4H.pdf

    I know for a), I can show it converges with integral test, but how can I find a specific partial sum?

    Do I just take the integral from 2 to n of 1/(x*(ln x)^2) and set it <10^-5 so that I get 5 digit accuracy, and then solve for N?

    For b), I know how to show divergence, but how do I find the specific partial sum? Same idea?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    I don't know exactly what resources you can use but for example if an alternating series u_1-u_2+u_3-\ldots satisfies the hypothesis of the Leibnitz's rule test then, |S_n-S|\leq u_{n+1} .
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by twittytwitter View Post
    http://math.rutgers.edu/~greenfie/we...fstuff/w4H.pdf

    I know for a), I can show it converges with integral test, but how can I find a specific partial sum?

    Do I just take the integral from 2 to n of 1/(x*(ln x)^2) and set it <10^-5 so that I get 5 digit accuracy, and then solve for N?

    For b), I know how to show divergence, but how do I find the specific partial sum? Same idea?
    a) the series...

    \displaystyle \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n\ \ln^{2} n} (1)

    ... is 'alternate sign' and the absolute value of the genral term decreases monotonically and tends to 0 with n, so that the series converges. For this type of series the truncation error is inferior, in absolute value, to the first 'discharged term' and that happens for...

    \displaystyle n\ \ln^{2} n \sim 10^{5} \implies n \sim 1800 (2)

    Kind regards

    \chi \sigma
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by twittytwitter View Post
    http://math.rutgers.edu/~greenfie/we...fstuff/w4H.pdf

    I know for a), I can show it converges with integral test, but how can I find a specific partial sum?

    Do I just take the integral from 2 to n of 1/(x*(ln x)^2) and set it <10^-5 so that I get 5 digit accuracy, and then solve for N?

    For b), I know how to show divergence, but how do I find the specific partial sum? Same idea?
    b) the series...

    \displaystyle \sum_{n=2}^{\infty} \frac{1}{n\ \sqrt{\ln n}} (2)

    ... diverges because is...

    \displaystyle I(x)= \int_{2}^{x} \frac{dt}{t\ \sqrt{\ln t}} = 2\ \sqrt {\ln x} - 2\ \sqrt{\ln 2} (2)

    ... and I(x) tends to infinity with x. The condition I(x)= 100 is verified for x \sim 5.5\ 10^{1085} ...

    Kind regards

    \chi \sigma
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  5. #5
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    Ok thanks, but that part for b) isn't what I had trouble with. Obviously if you take something crazy like 5.5*10^1085, it will work, but how can I find an n for which it will work and the partial sum will exceed 100 using paper and pencil (i.e. no computer programs).
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