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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.
That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
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 ...
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Similarly, [1] [()] (′ ( [])) [] = (′ ()) (″ ()) The above is obtained using a second order approximation, following the method used in estimating ...
All terms in a naïve Taylor expansion are identically zero. This is because the function / possesses an essential singularity at = in the complex -plane, and therefore the function is most appropriately modeled by a Laurent series-- a Taylor series has a zero radius of convergence. Thus, if a physical problem possesses a solution of this ...
Polynomial expansions such as the Taylor series expansion are often convenient for theoretical work but less useful for practical applications. Truncated Chebyshev series, however, closely approximate the minimax polynomial. One popular minimax approximation algorithm is the Remez algorithm.
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