Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The sine and cosine power series converge absolutely for every real argument

Statement

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

Facts & Assumptions

Given: A real xx.

[L1]

The sine and cosine series have terms x2n+1/(2n+1)!x^{2n+1}/(2n+1)! and x2n/(2n)!x^{2n}/(2n)! 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⁡∣ak+1/ak∣<1\limsup |a_{k+1}/a_k| < 1 gives absolute convergence and hence convergence, and lim inf⁡∣ak+1/ak∣>1\liminf |a_{k+1}/a_k| > 1 gives divergence).

Proof

technique · direct
1.1

For the sine absolute terms, the successive ratio is ∣x∣2/((2n+2)(2n+3))|x|^2/((2n+2)(2n+3)), which tends to 00.

L1algebra
1.2

For the cosine absolute terms, the successive ratio is ∣x∣2/((2n+1)(2n+2))|x|^2/((2n+1)(2n+2)), which tends to 00.

L1algebra
2.1

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

step 1.1step 1.2L2∎

Depends on

Used by

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

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Sources