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A power series is a type of series with terms involving a variable. More specifically, if the variable is \(x\), then all the terms of the series involve powers of \(x\). As a result, a power series can be thought of as an infinite polynomial.
In mathematics, a power series (in one variable) is an infinite series of the form = = + + + … where a n represents the coefficient of the nth term and c is a constant called the center of the series.
A power series about a, or just power series, is any series that can be written in the form, \[\sum\limits_{n = 0}^\infty {{c_n}{{\left( {x - a} \right)}^n}} \] where \(a\) and \({c_n}\) are numbers. The \({c_n}\)’s are often called the coefficients of the series. The first thing to notice about a power series is that it is a function of \(x\).
A power series is a series with terms involving a variable. More specifically, if the variable is \ (x\), then all the terms of the series involve powers of \ (x\). As a result, a power series can be thought of as an infinite polynomial. Power series are used to represent common functions and also to define new functions.
Power series are used to approximate functions that are difficult to calculate exactly, such as tan -1 (x) and sin (x), using an infinite series of polynomials. Power series are often used to approximate important quantities and functions such as π, e, and , an important function in statistics.
What is a power series? The power series, centered at c, is a series represented by the general form shown below. ∑ n = 0 ∞ a n (x − c) n = a 0 + a 1 (x − c) + a 2 (x − c) 2 + … The constants a n, where n ≥ 0, are called the series’ coefficients and c represents the center.
A power series in a variable is an infinite sum of the form. where are integers, real numbers, complex numbers, or any other quantities of a given type.