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There are a number of ways to demonstrate the validity of the small-angle approximations. 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 ...
Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts.
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines , each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines ...
The Taylor series of any polynomial is the polynomial itself.. The Maclaurin series of 1 / 1 − x is the geometric series + + + +. So, by substituting x for 1 − x, the Taylor series of 1 / x at a = 1 is
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.
This means that the radius of convergence of the Maclaurin series is (/) and the series will not converge for values of larger than this. The series can also be used for the hyperbolic case, in which case the radius of convergence is cos − 1 ( 1 / e ) − e 2 − 1 . {\displaystyle \cos ^{-1}(1/e)-{\sqrt {e^{2}-1}}.}
Machin-like formulas for π can be constructed by finding a set of integers , =, where all the prime factorisations of + , taken together, use a number of distinct primes , and then using either linear algebra or the LLL basis-reduction algorithm to construct linear combinations of arctangents of . For example, in the Størmer formula ...
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] Colin Maclaurin (1698–1746) Maclaurin also made significant contributions to the gravitation attraction of ellipsoids, a subject that furthermore attracted the attention of d'Alembert, A.-C ...