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Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

The Taylor expansion is the power series expansion of a function at a point. It represents a function as an infinite sum with terms calculated from the function's derivatives at that point. More precisely, It is defined as $$ f(x)\overset{x\to a}{=}\sum^{\infty}_{n=0}\frac{f^{(n)}(a)}{n!}(x-a)^n=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots $$

It happens often in applications that the Taylor expansion of $f$ at $a$ converges to $f$ (pointwise and locally uniformly) on some neighborhood of $a$: when this happens, the function is said to be analytic at $a$.

Applications:

A Taylor series is an idea used in computer science, calculus, and other kinds of higher-level mathematics. It is a series that is used to create an estimate (guess) of what a function looks like. Taylor Series are also used in power flow analysis of electrical power systems (Newton-Raphson method). Multivariate Taylor series is used in different optimization techniques; that is, you approximate your function as a series of linear or quadratic forms, and then successively iterate on them to find the optimal value.

References:

https://en.wikipedia.org/wiki/Taylor_series