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Math Help - problem with series limit..

  1. #1
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    problem with series limit..

    \lim_{n\to\infty}\sum_{n=1}^{\infty}\frac{1}{\sqrt  {n^2+n}}=?

    i tried to look for collapsing fractions but they don't collapse

    i tried to find the limit of the ratio of

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

    but i don't know how to begin untangling that thing

    is there another way or can you tell me how to begin to figure something out of that?
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  2. #2
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    Re: problem with series limit..

    You have a variable scope problem in your problem statement. Once the sum has occurred, there is no n left in the expression. Thus, the limit basically does nothing. Did you mean to use a dummy variable for the sum, and sum, say, from j = 1 to n?
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  3. #3
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    Re: problem with series limit..

    Quote Originally Posted by idom87 View Post
    \sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+n}}=?
    I assume you mean to ask about series convergence.

    Note that \frac{1}{\sqrt{n^2+n}}\ge\frac{1}{\sqrt{2n^2}}.

    What can be said of \sum_{n=1}^{\infty}\frac{1}{\sqrt{2}n}=?
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  4. #4
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    Re: problem with series limit..

    i don't know, i can't decompose the fraction
    what do you do in that case?
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  5. #5
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    Re: problem with series limit..

    Quote Originally Posted by idom87 View Post
    i don't know, i can't decompose the fraction
    what do you do in that case?
    First of all, is the problem as I changed correct?
    It is a series and not a limit.
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  6. #6
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    Re: problem with series limit..

    it is correct and

    \sum_{n=1}^{\infty}\frac{1}{\sqrt{2}n}\le\sum_{n=1  }^{\infty}\frac{1}{\sqrt{n^2+n}}
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  7. #7
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    Re: problem with series limit..

    Quote Originally Posted by idom87 View Post
    it is correct and

    \sum_{n=1}^{\infty}\frac{1}{\sqrt{2}~n}\le\sum_{n=  1}^{\infty}\frac{1}{\sqrt{n^2+n}}
    What can you say about the series \sum_{n=1}^{\infty}\frac{1}{n}~?
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  8. #8
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    Re: problem with series limit..

    \sum_{n=1}^{\infty}\frac{1}{\sqrt{2}n}\le\sum_{n=1  }^{\infty}\frac{1}{\sqrt{n^2+n}}\le\sum_{n=1}^{ \infty }\frac{1}{n}

    i assume your'e refereing to the sandwich rule but i need to find the limits of the larger and smaller sums and i don't know how
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  9. #9
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    Re: problem with series limit..

    Quote Originally Posted by idom87 View Post
    \sum_{n=1}^{\infty}\frac{1}{\sqrt{2}n}\le\sum_{n=1  }^{\infty}\frac{1}{\sqrt{n^2+n}}\le\sum_{n=1}^{ \infty }\frac{1}{n}
    i assume your'e refereing to the sandwich rule but i need to find the limits of the larger and smaller sums and i don't know how
    Why have you been asked to discuses series convergence when it is perfectly clear that you are missing the very basic ideas??


    Has your text/instructor discussed the harmonic series?

    If so what are the basics there?
    How does that apply here?

    If not, then you are not ready to do this question.
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  10. #10
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    Re: problem with series limit..

    ok i read about it in wikipedia and now i know and understand that \sum_{n=1}^{ \infty }\frac{1}{n}\to\infty

    what next?
    all of them \to\infty?
    Last edited by idom87; September 22nd 2011 at 02:10 PM.
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  11. #11
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    Re: problem with series limit..

    Quote Originally Posted by idom87 View Post
    ok i read about it in wikipedia and now i know and understand that \sum_{n=1}^{ \infty }\frac{1}{n}\to\infty

    what next?
    all of them \to\infty?
    Then any multiple of that also diverges.
    \sum {\frac{1}{{\sqrt 2 n}}} diverges.
    Use basic comparison tests.
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  12. #12
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    Re: problem with series limit..

    so the answer is that the series diverges?
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  13. #13
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    Re: problem with series limit..

    Quote Originally Posted by idom87 View Post
    so the answer is that the series diverges?
    Please answer the WHY question in reply #9.
    Do you have text material on series?
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  14. #14
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    Re: problem with series limit..

    i haven't been asked to do anything, i'm doing this on my own to prepare for the university
    i have a book for calculus1 and i have recorded video lectures, but i guess it doesn't cover everything because there was nothing specific about harmonic series either in the book or the lectures.. do you know anything online that you can recommend on the subject?
    how much am i missing here?
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