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The function e (−1/x 2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not. If f ( x ) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region.
The Bernoulli polynomials may be defined recursively by B 0 (x) = 1 and, for k ≥ 1, ′ = (), = The periodized Bernoulli functions are defined as = (⌊ ⌋), where ⌊x⌋ denotes the largest integer less than or equal to x, so that x − ⌊x⌋ always lies in the interval [0,1).
A Taylor series is a power series based on a function's derivatives at a single point. [3] More specifically, if a function f : U → R {\displaystyle f:U\to \mathbb {R} } is infinitely differentiable around a point x 0 {\displaystyle x_{0}} , then the Taylor series of f around this point is given by
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.
Example 2: The power series for g(z) = −ln(1 − z), expanded around z = 0, which is =, has radius of convergence 1, and diverges for z = 1 but converges for all other points on the boundary. The function f(z) of Example 1 is the derivative of g(z). Example 3: The power series
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.
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
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 = + + +.