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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 ...
Proof of lemma. The function () = (/) is the uniform limit of its Taylor expansion, which starts with degree 3. Also, ‖ ‖ <.Thus to -approximate () = using a polynomial with lowest degree 3, we do so for () with < / by truncating its Taylor expansion.
Its Taylor series about 0 is given by = (). The root test shows that its radius of convergence is 1. In accordance with this, the function f(z) has singularities at ±i, which are at a distance 1 from 0. For a proof of this theorem, see analyticity of holomorphic functions.
The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
Therefore, the radius of convergence of the Taylor series cannot be smaller than the distance from to the nearest singularity (nor can it be larger, since power series have no singularities in the interiors of their circles of convergence). A special case of the identity theorem follows from the preceding remark.
Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center c is equal to zero, for instance for Maclaurin series.
Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine, or on the differential equation ″ + = to which they are solutions. Elementary trigonometric identities