# Riemann summation method

A method for summing series of numbers. A series $\sum_{n=0}^\infty a_n$ can be summed by Riemann's method to a number $S$ if

$$\lim_{h\to0}\left[a_0+\sum_{n=1}^\infty a_n\left(\frac{\sin nh}{nh}\right)^2\right]=S.$$

This method was first introduced and its regularity was first proved by B. Riemann in 1854 (see ). The Riemann summation method has been applied in the theory of trigonometric series, where it is usually stated as follows: A trigonometric series

$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx$$

with bounded coefficients $a_n,b_n$ can be summed by Riemann's method at a point $x_0$ to a number $S$ if the function

$$F(x)=\frac{a_0x^2}{4}-\sum_{n=1}^\infty\frac{a_n\cos nx+b_n\sin nx}{n^2}$$

has, at $x_0$, Riemann derivative equal to $S$.

How to Cite This Entry:
Riemann summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_summation_method&oldid=33493
This article was adapted from an original article by T.P. Lykashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article