## Section8.1Definition 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{8.1.1}\\ \amp = c_0 + c_1(z-a) + c_2(z-a)^2 + c_3(z-a)^3 + \dots\tag{8.1.2} \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 (8.1.1), $$z$$ 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 8.4.

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 8.2. However, you should also gradually accumulate as many short-cut strategies as you can for finding these coefficients. The theorems in Section 8.10 will help.