How do you express, say, \frac{x+1}{x-1} as a sum of (positive and negative) powers?
Actually, come to think of it, I’m pretty sure you can’t: a_ix^i will always have its pole (if any) at x=0, and thus so will any sum of such terms, but a general rational function can have a pole anywhere.
\begin{align}
\dfrac{x+1}{x-1} = 1+\frac{2}{x-1} &= 1-2(1+x+x^2+\cdots) \\
&= -1-2x-2x^2-2x^3-\cdots
\end{align}
no negative powers
no; see above
NB re. bigger/smaller: two different series can easily have the same valuation: e.g. 1+x and 1+x^2 both start in degree 0
Yes, don’t forget that convergence is not required (that’s why they’re called formal series. You start to need negative powers of x only if you invert a power series whose constant term is 0.
Ah, OK, I misread: I thought you were specifying a finite number of terms, total, not just a finite number of negative-power terms. You can get poles just fine from an infinite number of positive-power terms, as @DPRK illustrated.
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