Complex analysis taylor

What is the Taylor's theorem analysis?
Recall that, in real analysis, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. f(x)≈f(a)+f′(a)(x−a).
This linear approximation fits f(x) with a line through x=a that matches the slope of f at a..
Why do we use Taylor series?
Taylor series expansion is an awesome concept, not only the world of mathematics, but also in optimization theory, function approximation and machine learning.
It is widely applied in numerical computations when estimates of a function's values at different points are required..
Recall that, in real analysis, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. f(x)≈f(a)+f′(a)(x−a).
This linear approximation fits f(x) with a line through x=a that matches the slope of f at a.

What is the Taylor's theorem analysis?