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In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann . One very common application is in numerical integration , i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule .
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets. Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent.
is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b. Any series that is not convergent is said to be divergent or to diverge.
The convergence of a geometric series can be described depending on the value of a common ratio, see § Convergence of the series and its proof. Grandi's series is an example of a divergent series that can be expressed as 1 − 1 + 1 − 1 + ⋯ {\displaystyle 1-1+1-1+\cdots } , where the initial term is 1 {\displaystyle 1} and the common ratio ...
The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral
Absolutely convergent series are unconditionally convergent. But the Riemann series theorem states that conditionally convergent series can be rearranged to create arbitrary convergence. [4] Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The general principle is that addition of infinite sums is ...
The same definition can be used for series = whose terms are not numbers but rather elements of an arbitrary abelian topological group.In that case, instead of using the absolute value, the definition requires the group to have a norm, which is a positive real-valued function ‖ ‖: + on an abelian group (written additively, with identity element 0) such that: