Maclaurin series and approximation

Build the standard series · Diagrams

Compare cosine with successive even-power partial sums

Build cosine with even powers

Use radians. Compare the constant, quadratic, quartic and sixth-degree partial sums with cosine, then move away from zero to inspect the error.

Series and source notes

P₀ = 1; P₂ = 1 − x²/2!; P₄ = P₂ + x⁴/4!; P₆ = P₄ − x⁶/6!. Odd-power coefficients vanish, so P₁ = P₀, P₃ = P₂, and P₅ = P₄.

The cosine series converges for every real x in radians. Shading shows a local comparison window, not a convergence interval or accuracy guarantee. A fixed truncation can have a large error far from zero.

The original sheet adds several partial sums before isolating P₆. This explorer compares one chosen truncation with cosine. Original selections and colours remain in the linked source diagram. The initial window is illustrative; the vertical range expands to include the selected values. Visual agreement near zero is not a convergence proof.

Five reconstructed stages compare blue cosine with the constant 1, then quadratic, quartic and sixth-degree partial sums; the last stage hides earlier polynomials. Source colours and selections are retained, including the constant changing from red to magenta. Illustrative common window minus 7 to 7 horizontally and minus 3 to 3 vertically; local agreement does not prove convergence.

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