ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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A rational function with nonvanishing denominator is locally represented by geometric-series expansions

Statement

The rational function $r(x)=1/(2-x)$ is real analytic on $\mathbb R\setminus\{2\}$ . At every $c\ne2$ it has the local expansion

$\frac1{2-x}=\sum_{n=0}^{\infty}\frac{(x-c)^n}{(2-c)^{n+1}}\qquad(|x-c|<|2-c|).$

Facts & Assumptions

Given: A centre $c\ne2$ .

[L1]

The geometric series converges for arguments of absolute value below $1$ (For $|r| < 1$ , $\sum_{k \ge 0} r^k = 1/(1-r)$ , and for $|r| \ge 1$ the series diverges).

[L2]

A nonvanishing analytic denominator has a local analytic reciprocal (A convergent real power series with nonzero constant term has a convergent reciprocal power series on a smaller neighbourhood, Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero).

Verification

technique · direct

1.1

Factor $2-x=(2-c)(1-(x-c)/(2-c))$ and apply [L1]. The resulting series is exactly the displayed one and converges when $|x-c|<|2-c|$ .

givenL1algebra

2.1

Since every $c\ne2$ admits this positive-radius local representation, $r$ is real analytic on its domain, in agreement with [L2].

step 1.1L2∎

Depends on

Used by

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Dependency tree · next 3 levels

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Sources

Analytic function, Encyclopedia of Mathematics (standard reference, not scraped)
Power series, Encyclopedia of Mathematics (standard reference, not scraped)
Northwestern Math 320-2 lecture notes (standard reference, not scraped)