Maclaurin series in the context of Infinite sequence

In mathematics, a power series (in one variable) is an infinite series of the form$${\displaystyle \sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\dots }$$where ${\displaystyle a_{n}}$ represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.

In many situations, the center c is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form$${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\dots .}$$

In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).

The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions.