Taylor Series in a sentence

Alternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series.

A function may not be equal to its Taylor series, even if its Taylor series converges at every point.

Calculation of Taylor series Several methods exist for the calculation of Taylor series of a large number of functions.

The Taylor series of a function is the limit of that function's Taylor polynomials as the degree increases, provided that the limit exists.

A function that is equal to its Taylor series in an open interval (or a disc in the complex plane ) is known as an analytic function in that interval.

And even if the Taylor series of a function f does converge, its limit need not in general be equal to the value of the function f(x).

Applying the multi-index notation the Taylor series for several variables becomes : which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, again in full analogy to the single variable case.

As a result, the radius of convergence of a Taylor series can be zero.

Both are defined via Taylor series analogous to the real case.

However, f(x) is not the zero function, so does not equal its Taylor series around the origin.

However, one may equally well define an analytic function by its Taylor series.

In general, Taylor series need not be convergent at all.

One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern.

Particularly convenient is the use of computer algebra systems to calculate Taylor series.

Taylor series are used to define functions and " operators " in diverse areas of mathematics.

The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series (analytic).

Then the Taylor series of f converges uniformly to some analytic function : Here comes the subtle point.

These are also the numbers appearing in the Taylor series expansion of tan(x) and tanh(x).

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.

The Taylor series expresses the function as a sum obtained from its derivatives at one point.