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Math Help - Tricky infinite series becomes Riemann Integral

  1. #1
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    Tricky infinite series becomes Riemann Integral

    Hey guys, I know this is a weird question but here it goes.. I few years ago I remember seeing an infinite series of the form \sum_{i=1}^\infty g(i) which on first glance appeared to diverge but it actually converged when you used the fact below


    If you can write \sum_{i=1}^\infty g(i) as \lim_{n\rightarrow\infty}\sum_{i=1}^n f\Big (\frac{i}{n}\Big )\frac{1}{n} then we know \sum_{i=1}^\infty g(i)=\int_0^1 f(x)dx

    Essentially I am looking for an example which appears to diverge, but can be turned into a Riemann integral which can be calculated to prove convergence.

    I am having a very hard time trying to remember/create such an example. Does anyone know of an example like this? If it's not clear what I'm talking about, the general version of the "fact" I mention can be found here as the corollary at the bottom [HTML]http://www.math.nus.edu.sg/~matngtb/Calculus/Riemannsum/Riemannsum.htm[/HTML]

    Thanks for all help
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  2. #2
    Member sbhatnagar's Avatar
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    Lightbulb Re: Tricky infinite series becomes Riemann Integral

    Consider the following 2 examples:

    Example 1

    \\\lim_{n\to \infty}(\frac{1}{1+n^3}+\frac{4}{8+n^3}+...+\frac{  1}{2n})\\\\=\lim_{n\to\infty}\sum_{r=1}^{n}\frac{1  }{n}(\frac{1}{\frac{r}{n}+\frac{n^2}{r^2}})\\\\= \int_{0}^{1} \frac{x^2}{x^3+1}dx

    Example 2

    \\\lim_{n\to\infty }(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+...+\frac{n}  {n^2+n^2})\\\\=\lim_{n\to\infty}\sum_{r=1}^{n} \frac{1}{n} (\frac{1}{1+\frac{r^2}{n^2}})\\\\=\int_{0}^{1} \frac{1}{x^2+1} dx



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    Last edited by sbhatnagar; November 5th 2011 at 07:41 AM. Reason: typo
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  3. #3
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    Re: Tricky infinite series becomes Riemann Integral

    thanks that is really close to what I remember but I could've sworn that the first step in each example could be written as the infinite series of a function of one variable. I could be remembering incorrectly but does anyone else have an example like this?
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    Member sbhatnagar's Avatar
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    Re: Tricky infinite series becomes Riemann Integral

    Quote Originally Posted by MonroeYoder View Post
    thanks that is really close to what I remember but I could've sworn that the first step in each example could be written as the infinite series of a function of one variable. I could be remembering incorrectly but does anyone else have an example like this?
    What do you mean by one variable?
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  5. #5
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    Re: Tricky infinite series becomes Riemann Integral

    your initial series involves a parameter. What I think I remember (and again I could definitely be wrong) is that some series \sum_{x=1}^\infty g(x) where g(x) is only a function of x. In other words, there are no other parameters like i or n, just a function consisting of fixed numbers and x like g(x) = \frac{x+1}{x+x^2}

    Then I "remember" letting x=\frac{i}{n} and getting it in a form where I could turn it into an integral.

    What I am "remembering" is an extra credit problem from a calc class I took. It was extra credit because just looking at \sum_{x=1}^\infty g(x) you thought the series would diverge since it sort of looked like the harmonic series but since you could convert it into an integral and you could show it actually converged.

    thanks again - it's possible I'm just crazy
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