In mathematics, a partition of an interval [a, b] on the real line is a finite sequence of the form a = x0 < x1 < x2 < ... < xn = b.
Such partitions are used in the theory of the Riemann integral and the Riemann-Stieltjes integral.
The mesh of the partition x0 < x1 < x2 < ... < xn is the length of the longest of these subintervals; it is max{ |xi − xi−1| : i = 1, ..., n }. As the mesh approaches zero, a Riemann sum based on the partition approaches the Riemann integral.
Then P is a sequence of partitions of the interval [a,b] and m is a sequence of non-negative real numbers.
Such a sequence r of Riemann sums for a function a defined on aninterval [a,b] is a sequence of real numbers and it may or may not converge.
For some functions a defined on aninterval [a,b], the corresponding sequence r of Riemann sums converges whenever the sequence P of partitions of the interval [a,b] is fine.
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