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In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum.It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus.
In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test .
In early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for , , , (the integral of ), (+) (the integral of sec, the inverse Gudermannian function), (), and (the Gudermannian function). However, thinking that he had merely redeveloped a method by Newton ...
For any real x, Newton's method can be used to compute erfi −1 x, and for −1 ≤ x ≤ 1, the following Maclaurin series converges: = = + +, where c k is defined as above. Asymptotic expansion
The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886. [1] The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin ...
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") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule".
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
The Maclaurin series of the logarithm function (+) is conditionally convergent for x = 1. The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.