Definition of Power Series

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Section 1.1 Definition of Power Series

Form of Power Series.

Most functions can be represented as a power series, whose general form is given by:

\begin{align} f(z) \amp = \sum_{n=0}^{\infty} c_n(z-a)^n\notag\\ \amp = c_0 + c_1(z-a) + c_2(z-a)^2 + c_3(z-a)^3 + \dots\tag{1.1.1} \end{align}

In this equation, for a given function \(f(z)\text{,}\) \(a\) is a constant that you get to choose and the \(c_n\)’s are constants that will be different if you change \(a\text{.}\)

Notation 1. Power Series.

In Equation (1.1.1), \(z\) is the independent variable of the function, \(a\) represents the point “around” which the function is being expanded, each of the constants \(c_n\) is called the coefficient of the \(n\)th term, and the entire \(n\)th term, i.e. \(c_n(z-a)^n\text{,}\) is called the \(n\)th order term. For further information about the geometric meaning of these new vocabulary words, see Section 1.4.

Informal theorem: A deep mathematical theorem guarantees that for each sufficiently smooth function \(f(z)\) and point \(a\text{,}\) the coefficients \(c_n\) are unique.

You can always find the coefficients, using the general method in Section 1.2. However, you should also gradually accumulate as many short-cut strategies as you can for finding these coefficients. The theorems in Section 1.10 will help.