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.. (Headbang)

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- Dec 7th 2012, 10:45 PMJDSConstruct 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.. (Headbang) - Dec 7th 2012, 11:34 PMMarkFLRe: Construct a table of Riemann sums to show that...
Let's look at the left sum first:

and so we have:

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

Now, to compute the limit:

We have the indeterminate form so applying L'Hôpital's rule, we find:

Now we have the indeterminate form , so applying L'Hôpital's rule again:

Now, see if you can do the same for the remaining two sums. - Dec 8th 2012, 01:49 PMJDSRe: 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

- Dec 8th 2012, 07:59 PMMarkFLRe: Construct a table of Riemann sums to show that...
Before constructing the table, can you compute the other two sums?

- Dec 9th 2012, 11:07 AMJDSRe: 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!

- Dec 12th 2012, 12:19 PMJDSRe: 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) - Dec 16th 2012, 02:44 PMJDSRe: Construct a table of Riemann sums to show that...
well, I am not sure what I am supposed to do here....any Ideas/Suggestions?