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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.
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 .
A difference engine is an automatic mechanical calculator designed to tabulate polynomial functions. It was designed in the 1820s, and was first created by Charles Babbage . The name difference engine is derived from the method of divided differences , a way to interpolate or tabulate functions by using a small set of polynomial co-efficients.
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]
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
The Maclaurin series of an even function includes only even powers. The Maclaurin series of an odd function includes only odd powers. The Fourier series of a periodic even function includes only cosine terms. The Fourier series of a periodic odd function includes only sine terms. The Fourier transform of a purely real-valued even function is ...
The Maclaurin series of the logarithm function (+) is conditionally convergent for x = 1. The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
for every ε > 0, and whether the corresponding series of the f(n) still diverges. Once such a sequence is found, a similar question can be asked with f(n) taking the role of 1/n, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series.