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Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm.
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.
Nonelementary antiderivatives can often be evaluated using Taylor series. Even if a function has no elementary antiderivative, its Taylor series can always be integrated term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius of convergence. However, even if the integrand has a convergent ...
Typical examples of analytic functions are The following elementary functions: All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series.
Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the ...
We derive Itô's lemma by expanding a Taylor series and applying the rules of stochastic calculus. Suppose is an Itô drift-diffusion process that satisfies the stochastic differential equation = +, where B t is a Wiener process.
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.