I'm in my second course of Calculus, this week my professor defined formal power series as follows:
Definition 1 Let $a$ be a real number. A formal power series centered at $a$ is any expression of the form $$\sum_{n=0}^{\infty}c_n(x-a)^n$$ where $c_0, c_1, \dots$ is a sequence of real numbers.
In practice, we find sometimes power series that don't look like the expression in the above definition, for instance, the following power series: $$\sum_{n=1}^{\infty}\frac{x^n}{n}$$ $$\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{2^{2n}(n!)^2}$$
Sctrickly speaking, the above two series are not formal power series in the sense of Definition $1$, since in the first one the index starts with $1$ instead of $0$ and in the second one the exponent of $x$ is $2n$ instead of $n$. My question is why sometimes the formal power series we use in practice are not consistent with the formal power series that we define in texts? I mean, why not avoid this by defining formal power series such that the index of the series can start with a nonnegative integer?
Best Answer
I can understand that reconciling the professor's definition with the formal power series used in practice can be difficult since the definition doesn't explain how we manipulate them. For instance, when $a_{2n}=0$ for all $n$, it's unclear whether the two formal expressions
$$ \sum_{n=0}^{\infty}a_n x^n\qquad\text{and}\qquad\sum_{k=1}^{\infty}a_{2k-1} x^{2k-1} $$
should be considered the same. The lack of explanation on how to identify and manipulate formal power series is a gap in the professor's definition. (It's similar to how we can't determine whether the expressions $2+3$ and $5$ are equivalent without defining the rules for interpreting $2$, $3$, $5$, and $+$.)
However, let me assure you that formal power series can be defined in a consistent manner that allows for manipulation, much like we do with numbers or polynomials. Once you accept this fact, it's likely that your doubts will be resolved.
In fact, formal power series are typically defined as the limit of polynomials using a specific type of limiting procedure. This means that the familiar arithmetic rules for polynomials carry over to formal power series.