Is the sum a formal power series?

Question

Given a series

$$\sum_{i \ge 0} x^{a\cdot i}(x^{b\cdot i} - 1)$$ for $a, b \in \mathbb{N}$

Would this series be considered a formal power series?

Answer 1

It depends on what you mean by the question.

If you're referring to the formula itself, it is not in the form of a formal power series. Nor is it in any of the other closely related forms that we don't bother distinguishing from the correct form.

However, if you mean the value of the formula as it would be calculated in formal power series about $x=0$, then if $a > 0$ or $b=0$, the formula does have a value that is a formal power series.

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To clarify what I mean about "closely related forms", technically speaking,

$$ \sum_{i = 0}^{\infty} x^{2i} $$

is not in the form of a power series. However, when we want to talk about things the form of a power series, we conventionally interpret such a formula as if it were instead the formula

$$ \sum_{n = 0}^{\infty} a_n x^{n} $$

where the sequence of coefficients $a_n$ is given by

$$ a_n = \begin{cases} 1 & n \text{ even} \\ 0 & n \text{ odd} \end{cases}$$