- #1
sponsoredwalk
- 533
- 5
Okay I've seen how crazy Riemann sums can get in real analysis and I've noticed a heirarchy of notation.
The Stewart/Thomas etc... kinds of books use;
[tex]\lim_{x \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x[/tex]
Where;
[tex]\Delta x = \frac{b - a}{n} and x_i = a + i\Delta x[/tex]
Then the books like Apostol and Bartle's real analysis use;
[tex]\lim_{x \to \infty} \sum_{i=1}^{n} f(x_i) (x_i - x_i_-_1)[/tex]and what I'd like to know is how to calculate the (x_i - x_i_-_1) for some equation like;
f(x) = x² integrated from 2 to 8. I can do the Δx = (b - a)/n version fine but how do you work the newer notation?
in f(x_i) (x_i - x_i_-_1) I would assume f(x_i) would use any endpoint, i.e. the right endpoint being a + iΔx but how do you make sense of the (x_i - x_i_-_1)?
The Stewart/Thomas etc... kinds of books use;
[tex]\lim_{x \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x[/tex]
Where;
[tex]\Delta x = \frac{b - a}{n} and x_i = a + i\Delta x[/tex]
Then the books like Apostol and Bartle's real analysis use;
[tex]\lim_{x \to \infty} \sum_{i=1}^{n} f(x_i) (x_i - x_i_-_1)[/tex]and what I'd like to know is how to calculate the (x_i - x_i_-_1) for some equation like;
f(x) = x² integrated from 2 to 8. I can do the Δx = (b - a)/n version fine but how do you work the newer notation?
in f(x_i) (x_i - x_i_-_1) I would assume f(x_i) would use any endpoint, i.e. the right endpoint being a + iΔx but how do you make sense of the (x_i - x_i_-_1)?
