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Math Help - Construct a table of Riemann sums to show that...

  1. #1
    JDS
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    Construct a table of Riemann sums to show that...

    Construct a table of Riemann sums to show that sums with right-endpoint, midpoint, and left-endpoint evaluation all converge to the same value as n approaches infinity.

    f(x) = sin x, [0, π/2]

    Thanks in advance!

    Kind of lost here..
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    MHF Contributor MarkFL's Avatar
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    Re: Construct a table of Riemann sums to show that...

    Let's look at the left sum first:

    \Delta=\frac{\frac{\pi}{2}-0}{n}=\frac{\pi}{2n}

    and so we have:

    A_n=\frac{\pi}{2n}\sum_{k=0}^{n-1}\sin\left(k\cdot\frac{\pi}{2n} \right)=\frac{\pi}{2n}\cdot\frac{1}{2}\left(\cot \left(\frac{\pi}{4n} \right)-1 \right)=

    \frac{\pi\left(\cot\left(\frac{\pi}{4n} \right)-1 \right)}{4n}

    To compute the sum, I used an identity of Lagrange:

    \sum_{n=1}^N \sin n\theta & = \frac{1}{2}\cot\frac{\theta}{2}-\frac{\cos(N+\frac{1}{2})\theta}{2\sin\frac{1}{2} \theta}

    Now, to compute the limit:

    \lim_{n\to\infty}\left( \frac{\pi\left(\cot\left( \frac{\pi}{4n} \right)-1 \right)}{4n} \right)=

    \frac{\pi}{4}\lim_{n\to\infty} \left( \frac{ \cot \left( \frac{\pi}{4n} \right)-1}{n} \right)=L

    We have the indeterminate form \frac{\infty}{\infty} so applying L'H˘pital's rule, we find:

    L=\frac{\pi}{4}\lim_{n\to\infty} \left( \frac{-\csc^2\left( \frac{\pi}{4n} \right)\left(-\frac{\pi}{4n^2} \right)}{1} \right)=\left( \frac{\pi}{4} \right)^2\lim_{n\to\infty} \left( \frac{\frac{1}{n^2}}{\sin^2\left( \frac{\pi}{4n} \right)} \right)

    Now we have the indeterminate form \frac{0}{0}, so applying L'H˘pital's rule again:

    L=\frac{\pi}{4}\lim_{n\to\infty} \left( \frac{\frac{2}{n}}{\sin\left( \frac{\pi}{2n} \right)} \right)=\lim_{n\to\infty} \left( \frac{\frac{\pi}{2n}}{\sin\left( \frac{\pi}{2n} \right)} \right)=1

    Now, see if you can do the same for the remaining two sums.
    Last edited by MarkFL; December 7th 2012 at 11:39 PM. Reason: hit submit too soon
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  3. #3
    JDS
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    Re: Construct a table of Riemann sums to show that...

    Thanks for the reply, I got a bit busy so late getting back to this one. In the book, they give an example where it shows a table with values of n (that I am supposed to select) and the corresponding left endpoint, midpoint and right end point evaluation all converge to the same value as n approaches infinity. So, i know that they supposedly converge to 1 (based on the info you have so graciously shown), however, how do I show this in the table. I mean, I obviously know how to make a table, and of course I pick my own (ever increasing) n values, but what method am I supposed to use to then substitute "my" N values in and then solve for left-mid-right end points. (which then should show that they all converge to 1) I hope that makes sense! If not I can try and explain further
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Construct a table of Riemann sums to show that...

    Before constructing the table, can you compute the other two sums?
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  5. #5
    JDS
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    Re: Construct a table of Riemann sums to show that...

    oh, yeah, I think so....I will give it a shot, but I might not get to it today, Kind of a busy schedule but as soon as I get it done I will post here. Thanks in advance!
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  6. #6
    JDS
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    Re: Construct a table of Riemann sums to show that...

    Been crazy busy, but this is all I could manage to figure out....

    (see attached picture)
    Attached Thumbnails Attached Thumbnails Construct a table of Riemann sums to show that...-calculus-prob1.jpg  
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  7. #7
    JDS
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    Re: Construct a table of Riemann sums to show that...

    well, I am not sure what I am supposed to do here....any Ideas/Suggestions?
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