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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
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of 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.
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]
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}} .
In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion = + + +.
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.
Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series. The existence of some formal power series g(D) with nonzero constant coefficient, such that He n (x) = g(D)x n, is another equivalent to the statement that these polynomials form an Appell sequence.