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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 Taylor series of f converges uniformly to the zero function T f (x) = 0, which is analytic with all coefficients equal to zero. The function f is unequal to this Taylor series, and hence non-analytic. For any order k ∈ N and radius r > 0 there exists M k,r > 0 satisfying the remainder bound above.
Faà di Bruno's formula gives coefficients of the composition of two formal power series in terms of the coefficients of those two series. Equivalently, it is a formula for the nth derivative of a composite function. Lagrange reversion theorem for another theorem sometimes called the inversion theorem; Formal power series#The Lagrange inversion ...
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.
9 External links. Toggle the table of contents ... the series above equals the Taylor expansion of the geometric series and consequently , ... ISBN 0-12-558840-2. Kai ...
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
For given x, Padé approximants can be computed by Wynn's epsilon algorithm [2] and also other sequence transformations [3] from the partial sums = + + + + of the Taylor series of f, i.e., we have = ()!. f can also be a formal power series, and, hence, Padé approximants can also be applied to the summation of divergent series.