# Does “5 \< ∞“ make any sense?

**URL:** <https://boards.straightdope.com/t/does-5-make-any-sense/958250>\
**Category:** Factual Questions\
**Created:** [January 22, 2022, 11:13pm UTC](https://boards.straightdope.com/t/does-5-make-any-sense/958250 "2022-01-22T23:13:04Z")\
**Posts on this page:** 6\
**Page:** 2

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [January 26, 2022, 1:30am UTC](https://boards.straightdope.com/t/does-5-make-any-sense/958250/21 "2022-01-26T01:30:58Z")

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How do you express, say, \frac{x+1}{x-1} as a sum of (positive and negative) powers?

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [January 26, 2022, 1:39am UTC](https://boards.straightdope.com/t/does-5-make-any-sense/958250/22 "2022-01-26T01:39:40Z")

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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.

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**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [January 26, 2022, 6:15am UTC](https://boards.straightdope.com/t/does-5-make-any-sense/958250/23 "2022-01-26T06:15:12Z")

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> [@Chronos](#):
>
> How do you express, say, \frac{x+1}{x-1} as a sum of (positive and negative) powers?

\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

> [@Chronos](#):
>
> Actually, come to think of it, I’m pretty sure you can’t: a\_ix^i aixia\_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.

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

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [January 26, 2022, 3:34pm UTC](https://boards.straightdope.com/t/does-5-make-any-sense/958250/24 "2022-01-26T15:34:07Z")

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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.

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [January 26, 2022, 9:44pm UTC](https://boards.straightdope.com/t/does-5-make-any-sense/958250/25 "2022-01-26T21:44:16Z")

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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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**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\
**Post date:** [January 26, 2022, 11:02pm UTC](https://boards.straightdope.com/t/does-5-make-any-sense/958250/26 "2022-01-26T23:02:19Z")

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