Skip to main content Contents Index Prev Up Next \(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
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Section 13.1 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{13.1.1}
\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 13.1 . Power Series.
In
Equation (13.1.1) ,
\(z\) 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 13.4 .
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 13.2 . However, you should also gradually accumulate as many short-cut strategies as you can for finding these coefficients. The theorems in
Section 13.10 will help.
