Taylor series
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A Taylor series is a way to express a complicated function as an infinite sum of simpler terms, making it easier to work with mathematically. This matters because it allows mathematicians and scientists to approximate functions, solve equations, and perform calculations that would otherwise be difficult or impossible to do.
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As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.
In mathematical analysis, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.