I am struggling to understand what the order of a real formal power series is. I am working through the book "Topology: An Introduction" by Stefan Waldmann. There he describes the "order" of a formal power series as follows:
We consider the real formal power series $a = \sum_{n = 0}^\infty a_nx^n$ with coefficients $a_n \in \mathbb{R}$ and a variable $x$. We define now the order of $a$ to be
$$o(a) = \begin{cases} \infty & \text{if } a = 0 \\ \min\{n | a_n \neq 0\} & \text{if } a \neq 0 \\ \end{cases}$$
However, shouldn't it be "max" here instead of "min"? For example, if we have
$$a = 1 - 3x + 5x^2 - 7x^3 + 0x^4 + 0x^5 ...$$ $$b = 2 + 4x + 0x^2 + 0x^3 + 0x^4 + 0x^5 ...$$
They would both have an order of $0$. However if I would consider these as polynomials, I'd say they have orders $3$ and $1$ respectively.
It is correct. The order of $a$ is defined as the smallest power of $x$ with a nonzero coefficient. See here. This also fits to $o(0) = \infty$ because the minimum of the empty set is usually defined as $\infty$.
You must not confuse this with the degree of a polynomial. Power series which are no polynomials have infinitely many nonzero coefficients, thus their degree could be defined as $\infty$, but this is not an interesting information.