Definition of Power Series

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Section 7.1 Definition 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 \nonumber\tag{7.1.1}\\ \amp = c_0 + c_1(z-a) + c_2(z-a)^2 + c_3(z-a)^3 + \dots\tag{7.1.2} \end{align}

In this equation, for a give 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.

In Equation (7.1.1), \(z\) 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 7.4.

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 7.2. However, you should also gradually accumulate as many short-cut strategies as you can for finding these coefficients. The theorems in Section 7.10 will help.