Courses in Math - Taylor's Series - Definition

A Taylor's series means a series expansion of an analytic function.

Taylor's Series - Definition

When function f(z) is analytic in the neighbourhood of a point a embedded in the complex space it can be uniquely represented by the power series

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In that way we end up with the Taylor's series:

The function is called analytical at point a if it is differentiable everywhere around that point (i. e. at each point within the circle of small enough radius r and centered at the a point).
Sometimes could be useful to have a truncated Taylor's series e. g. in the Taylor Collocation Method. Let's consider the example of f(t) function approximated by a truncated Taylor's series:

where beta is a parameter in [0, 1] and dt denotes the time increment (t_n+1 - t_n). The upper dot means the first time derivative.