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Math Help - Dirichlet's Test on following sum

  1. #1
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    Dirichlet's Test on following sum

    Show that the series 1/(2log2)+1/(3log3)-1/(4log4)-1/(5log5)-1/(6log6)-1/(7log7)+.... is convergent, the rule of signs being that successive terms with the same sign come in groups of 2, 4, 8, 16... Begin by considering the series 1/2+1/3-1/4-1/5-1/6-1/7+1/8...

    What i know is that the sum of 1/logn goes to 0 and decreasing, so that part is fine, but I can't find a bound on the 1/2+1/3-1/4-1/5-1/6-1/7+1/8... to satisfy the Dirichlet Test condition.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    The sum you're talking about is  \sum_{n=2}^\infty \frac{(-1)^{\lfloor \log_2(n)\rfloor-1}}{n} , who's partial sums oscillate wildly between  0 and  1 . It's hard to tell for inspection whether this sum converges or not but it does appear to be bounded.

    What is this problem for? Is there any information you're leaving out?
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    I think the sum I just mentioned diverges. See here.

    The best way to show your original sum converges is by showing it converges absolutely. Can you take it from here?

    Edit: The series is not absolutely convergent. It fails the integral test. Sorry for the mistake!
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by jordanhe View Post
    Show that the series 1/(2log2)+1/(3log3)-1/(4log4)-1/(5log5)-1/(6log6)-1/(7log7)+.... is convergent, the rule of signs being that successive terms with the same sign come in groups of 2, 4, 8, 16... Begin by considering the series 1/2+1/3-1/4-1/5-1/6-1/7+1/8...

    What i know is that the sum of 1/logn goes to 0 and decreasing, so that part is fine, but I can't find a bound on the 1/2+1/3-1/4-1/5-1/6-1/7+1/8... to satisfy the Dirichlet Test condition.
    Lets write...

    \frac{1}{2} + \frac{1}{3} - \frac{1}{4} - \frac{1}{5} - \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \dots = \sum_{n=1}^{\infty} (-1)^{n+1} a_{n} (1)

    ... where...

    a_{n} = \frac{1}{2^{n}} + \frac{1}{2^{n}+1} + \dots + \frac{1}{2^{n+1}-1} (2)

    But a_{n}<1 and any partial sum \sum_{n=1}^{N} (-1)^{n+1} is bounded, so that also any partial sum \sum_{n=1}^{N} (-1)^{n+1} a_{n} is bounded...

    Kind regards

    \chi \sigma
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by chisigma View Post
    Lets write...

    \frac{1}{2} + \frac{1}{3} - \frac{1}{4} - \frac{1}{5} - \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \dots = \sum_{n=1}^{\infty} (-1)^{n+1} a_{n} (1)

    ... where...

    a_{n} = \frac{1}{2^{n}} + \frac{1}{2^{n}+1} + \dots + \frac{1}{2^{n+1}-1} (2)

    But a_{n}<1 and any partial sum \sum_{n=1}^{N} (-1)^{n+1} is bounded, so that also any partial sum \sum_{n=1}^{N} (-1)^{n+1} a_{n} is bounded...

    Kind regards

    \chi \sigma
    Here's a similar approach:

    If we look at the original sum and do what chisigma did we get the following:

    <br /> <br />
b_{n} = \frac{1}{2^{n}\log(2^n)} + \frac{1}{(2^{n}+1)\log(2^n+1)} + \dots + \frac{1}{(2^{n+1}-1)\log(2^{n+1}-1)} < \frac{a_n}{\log(2^n)} where  a_n is defined in the post above. Since  a_n<1 we get that  b_n\to0 so by the alternating series test our original sum converges since it equals  \sum (-1)^{n+1}b_n .
    Last edited by chiph588@; May 19th 2010 at 09:47 AM.
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  6. #6
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    I think I know how to do it now. The thing is that we need to do the Dirichlet's Test twice. First, u use that on 1/2+1/3-1/4....., since the hypo-geometric series goes to zero and also decreasing, and the other one (1,1,-1,-1,-1,-1...) always bounded by zero, so this series is convergent. And then, since this is convergent, and 1/logn is decreasing and going to zero, so the original series will also be convergent.
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  7. #7
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by jordanhe View Post
    I think I know how to do it now. The thing is that we need to do the Dirichlet's Test twice. First, u use that on 1/2+1/3-1/4....., since the hypo-geometric series goes to zero and also decreasing, and the other one (1,1,-1,-1,-1,-1...) always bounded by zero, so this series is convergent. And then, since this is convergent, and 1/logn is decreasing and going to zero, so the original series will also be convergent.
    That's the right idea, but keep in mind that first sum doesn't converge but is bounded and that's all we need.
    Last edited by chiph588@; May 20th 2010 at 06:37 PM.
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