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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.
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 extremely slow convergence of the arctangent series for | | makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed 1 4 π {\displaystyle {\tfrac {1}{4}}\pi } as a sum of arctangents of smaller values, eventually resulting in a variety of ...
The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments.
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.
where the power series on the right-hand side of is expressed in terms of the (generalized) binomial coefficients ():= () (+)!.Note that if α is a nonnegative integer n then the x n + 1 term and all later terms in the series are 0, since each contains a factor of (n − n).
If is expressed in radians: = = These limits both follow from the continuity of sin and cos. =. [7] [8] Or, in general, =, for a not equal to 0. = =, for b not equal to 0.
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 = + + +.