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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 .
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 ...
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 process of finding the value of a sum is called integration. In technical language, integral calculus studies a certain linear operator. The Riemann sum inputs a function and outputs a function, which gives the algebraic sum of areas between the part of the graph of the input and the x-axis.
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 .
Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. For a left continuous, locally bounded and adapted process H the integral H · X exists, and can be calculated as a limit of Riemann sums.
The main term on the left is Φ(1); which turns out to be the dominant terms of the prime number theorem, and the main correction is the sum over non-trivial zeros of the zeta function. (There is a minor technical problem in using this case, in that the function F does not satisfy the smoothness condition.)
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral: ∑ x < n ≤ y a n ϕ ( n ) = A ( y ) ϕ ( y ) − A ( x ) ϕ ( x ) − ∫ x y A ( u ) d ϕ ( u ) . {\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x ...