LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

The sine and cosine power series converge absolutely for every real argument

Statement

For every real $x$ , the defining power series of $\sin x$ and $\cos x$ converge absolutely. Equivalently, both have infinite radius of convergence.

Facts & Assumptions

Given: A real $x$ .

[L1]

The sine and cosine series have terms $x^{2 n + 1} / (2 n + 1)!$ and $x^{2 n} / (2 n)!$ up to signs (Sine and cosine defined by their real power series).

[L2]

The ratio test proves absolute convergence when the ratio of successive absolute terms tends to a limit less than one (Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence).

Proof

technique · direct

1.1

For the sine absolute terms, the successive ratio is $∣ x ∣^{2} / ((2 n + 2) (2 n + 3))$ , which tends to $0$ .

L1algebra

1.2

For the cosine absolute terms, the successive ratio is $∣ x ∣^{2} / ((2 n + 1) (2 n + 2))$ , which tends to $0$ .

L1algebra

2.1

The ratio test proves absolute convergence of both series for this arbitrary $x$ .

step 1.1step 1.2L2∎

Depends on

Used by

Hadamard instability despite analytic solvability Counterexample
The derivatives of sine and cosine are cosine and minus sine Theorem

Cited to discharge well-definedness by Sine and cosine defined by their real power series.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
C. Schmeiser, Introduction to Analysis (standard reference, not scraped)