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..
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..
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.
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