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For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.
It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by Brook Taylor, [8] after whom the series are now named. The Maclaurin series was named after Colin Maclaurin, a Scottish mathematician, who published a special case of the Taylor result in the mid-18th century.
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
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 .
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 ...
For example, using Cauchy's integral formula for any positively oriented Jordan curve which parametrizes the boundary of a region , one obtains expressions for the derivatives f (j) (c) as above, and modifying slightly the computation for T f (z) = f(z), one arrives at the exact formula
These two rules can be associated with Euler–MacLaurin formula with the first derivative term and named First order Euler–MacLaurin integration rules. [7] The two rules presented above differ only in the way how the first derivative at the region end is calculated.
The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation , cos θ {\displaystyle \textstyle \cos \theta } is approximated as either 1 {\displaystyle 1} or as 1 − 1 2 θ 2 {\textstyle 1-{\frac {1}{2}}\theta ^{2}} .