Question

Define the term power series.

Short answer

The explanation is stated below.

Step-by-step solution

Form of power series.

A power series is a series which is in the form of,

\(\sum\limits_{n = 0}^\infty {{c_n}} {x^n} = {c_0} + {c_1}x + {c_2}{x^2} + {c_3}{x^3} + \cdots \)

Explanation of power series.

Where, \(x\) is a variable and \({c_n}{\rm{'s }}\) are constant coefficient of the series.

A series is an infinite series centered either at\(x = 0{\rm{ or }}x = a\).

If the series is centered at\(x = 0{\rm{ or }}x = a\), then the series becomes an infinite series.

If the series is centered at\({C_0},C1,{C_2},, \cdots \), then the power series becomes,

\(x = 0{\rm{ }}\sum\limits_{n = 0}^\infty {{c_n}} {x^n} = {c_0} + {c_1}{x^2} + \cdots + {c_n}{x^n} + \cdots \)

If the series is centered at\({\rm{ }}x = a\), then the power series becomes,

\(\sum\limits_{n = 0}^\infty {{c_n}} {(x - a)^n} = {c_0} + {c_1}(x - a) + {c_2}(x - a) + \cdots + {c_n}{(x - a)^n} + \cdots \)

Where, \(a\)is a center and \({C_0},C1,{C_2},, \cdots \) are all constants.