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Loosely speaking, a function is Riemann integrable if all Riemann sums converge as the partition "gets finer and finer". While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using ...
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.
The harmonic number H n can be interpreted as a Riemann sum of the integral: + = (+). The n th harmonic number is about as large as the natural logarithm of n . The reason is that the sum is approximated by the integral ∫ 1 n 1 x d x , {\displaystyle \int _{1}^{n}{\frac {1}{x}}\,dx,} whose value is ln n .
Roughly speaking, one chooses a sequence of partitions of the interval from 0 to t and constructs Riemann sums. Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. It is crucial which point in each of the small intervals is used to compute the value of the function. The limit then is taken in ...
The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [ a , b ] {\displaystyle [a,b]} be a closed interval of the real line; then a tagged partition P {\displaystyle {\cal {P}}} of [ a , b ] {\displaystyle [a,b]} is a finite sequence
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...
Riemann sum; Riemann surface; Riemann xi function; Riemann zeta function; Riemann–Hilbert correspondence; Riemann–Hilbert problem; Riemann–Lebesgue lemma; Riemann–Liouville integral; Riemann–Roch theorem; Riemann–Roch theorem for smooth manifolds; Riemann–Siegel formula; Riemann–Siegel theta function; Riemann–Silberstein ...