Ads
related to: maclaurin series for common functions practice exercises worksheet
Search results
Results From The WOW.Com Content Network
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.
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.
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.
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.
for the infinite series. Note that if the function () is increasing, then the function () is decreasing and the above theorem applies.. Many textbooks require the function to be positive, [1] [2] [3] but this condition is not really necessary, since when is negative and decreasing both = and () diverge.
Differentiating term-wise the binomial series within the disk of convergence | x | < 1 and using formula , one has that the sum of the series is an analytic function solving the ordinary differential equation (1 + x)u′(x) − αu(x) = 0 with initial condition u(0) = 1. The unique solution of this problem is the function u(x) = (1 + x) α.
Maclaurin used Taylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. 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]
These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function . Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and P 0 ( x ) is not even a function, being the derivative of a ...