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t. e. In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.
The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives f(k)(x) evaluated at the endpoints of the interval, that is to say x = m and x = n. Explicitly, for p a positive integer and a function f(x) that is p times continuously differentiable on the interval [m,n ...
22.1 million (2015) [ 5 ] Deaths. 16,100 (2015) [ 6 ] Kidney stone disease, also known as renal calculus disease, nephrolithiasis or urolithiasis, is a crystallopathy where a solid piece of material (renal calculus) develops in the urinary tract. [ 2 ] Renal calculi typically form in the kidney and leave the body in the urine stream. [ 2 ]
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.
Maclaurin attributed the series to Brook Taylor, though the series was known before to Newton and Gregory, and in special cases to Madhava of Sangamagrama in fourteenth century India. [6] Nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as the Maclaurin series. [7 ...
t. e. 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.