DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Sine and cosine defined by their real power series

Definition

For $x \in R$ , define $\sin x : = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{2 n + 1}}{(2 n + 1)!}, \cos x : = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{2 n}}{(2 n)!} .$

These are real power series in the sense of A real power series about a centre, its interval of convergence, and its radius in $[0, + \infty]$ . Their convergence for every real argument is discharged by The sine and cosine power series converge absolutely for every real argument.

Sine and cosine defined by their real power series

Definition

Depends on

Dependency tree · 2 levels

Sources