Maclaurin series and approximation
Build the standard series · Diagrams · Complete reading
Compare the logarithm with successive partial sums
Build the logarithm one power at a time
Compare ln(1 + x) with one partial sum at a time. Choose a source degree, then move x and inspect the numerical error.
Series and source notes
P₀ = 0; P₁ = x; P₂ = x − x²/2; P₃ = x − x²/2 + x³/3; P₄ = x − x²/2 + x³/3 − x⁴/4.
The infinite series equals ln(1 + x) for −1 < x ≤ 1. At x = 1, the alternating harmonic series converges to ln 2. At x = −1, the function is undefined and the series diverges. For x > 1, the terms do not tend to zero. The shading marks only the open interval; its endpoints need separate checks.
The original sheet adds several polynomials before isolating P₄. This explorer compares one polynomial with the function. Original selections and colours remain in the linked source diagram. The initial window is illustrative, not an original calculator setting; the vertical range expands to include the selected values. Visual agreement is not a convergence proof.
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.