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Using Riemann Upper sums to solve limits

  1. Nov 5, 2007 #1

    Gib Z

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    I often see people use the Riemann definition of the integral to solve a certain limit-series computation, but they usually just skip a step that I can follow one way but not the other. Given the integral, I can see the limit-series that comes from it, but when trying to find the integral from the limit-series I have a problem.

    For example, over the interval a=0 and b=1, [tex]\int^1_0 f(x) dx = \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n}\right)[/tex] using Riemann Upper sums.

    So basically If required to somehow work out [tex]f(k/n) = something[/tex], so that I can re-express it in terms of f(x). So i guess the question is, does anyone know a general method to spot the function of k/n, as to find f(x) ?
  2. jcsd
  3. Nov 5, 2007 #2
    I dont believe is any, rather than pure inspection.

    In most examples I've seen, the sum can be worked to be on the most common form

    [tex]\lim_{\Delta P\rightarrow 0}\sum_{n=0}^\infty f(x_i^*)(x_{i+1}-x_i),[/tex]

    where [itex]P=\{x_0,...,x_n,...\}[/itex] and [itex]x_i^* \in (x_i,x_{i+1})[/itex], so its easier to spot the [itex]f[/itex].
  4. Nov 6, 2007 #3

    Gib Z

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    Damn I was hoping that wouldn't be the answer :( Thanks
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