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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.
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 ...
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.
These two rules can be associated with Euler–MacLaurin formula with the first derivative term and named First order Euler–MacLaurin integration rules. [8] The two rules presented above differ only in the way how the first derivative at the region end is calculated.
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Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
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.
Independently from Euler and using the same methods, Maclaurin discovered the Euler–Maclaurin formula. He used it to sum powers of arithmetic progressions, derive Stirling's formula, and to derive the Newton–Cotes numerical integration formulas which includes Simpson's rule as a special case. [7] Maclaurin contributed to the study of ...