Alphabeta MathFormal Power Series: Examples and Counterexamples
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✓ 9 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Number Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Roots, Rational Powers, and Classical Inequalities
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
2 · Logical flowchart
3 · Items
Example
The formal geometric identity (1-x)⁻¹=∑_n≥0xⁿ holds over every commutative ring
Example
Negative binomial series: (1-x)⁻ᵐ=∑_n≥0C(m+n-1, n)xⁿ for m≥1
Example
The constant-one square root of 1-4x and its first coefficients
Example
Lagrange inversion gives the Catalan coefficients of the inverse of x-x²
Example
The compositional inverse of x/(1-x) is x/(1+x)
Counterexample
x has no multiplicative inverse in R⟦ x⟧ although it is invertible in K((x)) when R=K is a field
Counterexample
Substituting 1 into 1+x+x²+⋯ is not a defined formal composition
Counterexample
An infinite family of constant series 1 is not summable in the formal topology
Example