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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
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
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.
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.
The case α = 1 gives the series 1 + x + x 2 + x 3 + ..., where the coefficient of each term of the series is simply 1. The case α = 2 gives the series 1 + 2x + 3x 2 + 4x 3 + ..., which has the counting numbers as coefficients. The case α = 3 gives the series 1 + 3x + 6x 2 + 10x 3 + ..., which has the triangle numbers as coefficients.
The Maclaurin series of the logarithm function (+) is conditionally convergent for x = 1 (see the Mercator series). 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.
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 = + + +.
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 ...