As we know, for $q \in \mathbb N$ we have $x^q - 1 = (x-1) \cdot (x^{q-1} + x^{q-2} + \cdots + 1)$.

Does some kind of similar expansion exist for $q \in \mathbb R_{>0}$? I'm not sure, but at the very least

$$\lim_{x \to 1} \frac {x^q - 1} {x - 1} = \lim_{x \to 1} q x^{q-1} = q,$$

so it would make some sense to separate $x-1$ from everything else.

My only thought is to use Taylor expansion for $(1 + t)^q$ (where $t = x - 1$):

$$\frac {(1 + t)^q - 1} {t},$$

and it'll find expression as a power series of $t$. But this power series

  • Doesn't always converge (while the initial expression is defined everywhere)
  • Is in terms of $t=x-1$. It's unclear to me how to convert it into power series of $x$ so that coefficients have a reasonable representation.

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