Kashmir
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I can't understand how this approximation works ##\sum_{k=0}^m\left(\frac{k}{m}\right)^n\approx\int_0^m\left(\frac{x}{m}\right)^ndx\tag{1}##Can you please help me
We first think about the sum as the area under the rectangles having unit width.andrewkirk said:Choose a moderate value of m, not too small, say m=20.
Then set n=1, and draw the graph of ##f(x) = (\frac xm)^n## between 0 and ##m##. On the same set of axes draw the series of m rectangles such that the k-th rectangle (k = 0, ..., m-1) is the set of points (x,y) with ##\frac km\le x\le \frac{k+1}m## and ##(\frac xm)^n\le y\le (\frac{x+1}m)^n##.
Then do the same thing on a new set of axes, for ##n=2##.
That should give you a strong sense for why the approximation works.
thank you so muchJFerreira said:Geogebra can help you see the graphs: https://www.geogebra.org/graphing/sfgvm5zu
Yes, that's correct. It is called a Riemann Sum. You can read more about them here, including some nice pictures.Kashmir said:We first think about the sum as the area under the rectangles having unit width.
This area is roughly the integral of the corresponding continuous function.
Am i right?
Thank you :)andrewkirk said:Yes, that's correct. It is called a Riemann Sum. You can read more about them here, including some nice pictures.