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Watch how polynomial approximations converge to transcendental functions, term by term. Explore Taylor and Maclaurin series with interactive controls and animated plots.
sin(x) — x − x³/3! + x⁵/5! − ⋯(R = ∞)
sin(x) vs T_1(x) — 1 term
Each term = f^(n)(a)/n! · (x-a)^n
|f(x) - T_n(x)| at sample points
The Taylor series of a function f(x) around a point a is defined as:
When a = 0 this is called a Maclaurin series. The approximation improves as more terms are added, and the series converges within the radius of convergence R of the function.
x − x³/3! + x⁵/5! − x⁷/7! + ⋯
R = ∞ (converges everywhere)
1 − x²/2! + x⁴/4! − x⁶/6! + ⋯
R = ∞ (converges everywhere)
1 + x + x²/2! + x³/3! + x⁴/4! + ⋯
R = ∞ (converges everywhere)
x − x²/2 + x³/3 − x⁴/4 + ⋯
R = 1 (converges for |x| < 1)
x − x³/3 + x⁵/5 − x⁷/7 + ⋯
R = 1 (converges for |x| ≤ 1)
1 + x + x² + x³ + x⁴ + ⋯
R = 1 (converges for |x| < 1)
x + x³/3! + x⁵/5! + x⁷/7! + ⋯
R = ∞ (converges everywhere)
1 + x²/2! + x⁴/4! + x⁶/6! + ⋯
R = ∞ (converges everywhere)
1 + x/2 − x²/8 + x³/16 − ⋯
R = 1 (binomial, n=½)
x + x³/3 + 2x⁵/15 + 17x⁷/315 + ⋯
R = π/2
1 + 2x + x² (exact, finite)
R = ∞ (polynomial)