Taylor & Maclaurin Series Calculator
Calculate the Maclaurin (Taylor about x=0) series expansion for common mathematical functions and evaluate polynomial approximations at any x value.
Taylor Series Guide
What Is a Taylor Series?
A Taylor series expresses a function as an infinite sum of polynomial terms. For a function f(x) about point a: f(x) = f(a) + f'(a)(x−a)/1! + f''(a)(x−a)²/2! + f'''(a)(x−a)³/3! + ... When a=0, this is the Maclaurin series. Common series: eˣ = 1 + x + x²/2! + x³/3! + ... (all x). sin(x) = x − x³/3! + x⁵/5! − ... (all x, x in radians). cos(x) = 1 − x²/2! + x⁴/4! − ... (all x, x in radians). ln(1+x) = x − x²/2 + x³/3 − ... (−1 < x ≤ 1). 1/(1−x) = 1 + x + x² + x³ + ... (|x| < 1).
Why Taylor Series Matter
Taylor series are one of the most powerful tools in mathematics, physics, and engineering. Approximation: for small x, sin(x) ≈ x (first term). This small-angle approximation is used throughout physics — pendulum motion, optics, relativistic corrections. Numerical computation: computers evaluate transcendental functions (sin, cos, exp, ln) using polynomial approximations derived from Taylor series — faster to compute than the exact function. Error estimation: the remainder term bounds how wrong
Convergence and Radius
A Taylor series only equals the original function within its radius of convergence. eˣ: converges for all x (radius = ∞). sin(x), cos(x): converges for all x. ln(1+x): converges for −1 < x ≤ 1. 1/(1−x): converges for |x| < 1. Outside the radius: the series diverges (partial sums grow without bound). At the boundary (x at the edge of radius): convergence may be conditional. Alternating series test: an alternating series (terms alternate in sign) converges if terms decrease monotonically to zero —
Applications in Physics
Small-angle approximation: sin(θ) ≈ θ for θ < 0.2 rad (~11°). Error < 1% for θ < 10°. Used in: simple harmonic motion equations, small-signal amplifier analysis, thin lens formula derivation. Binomial series: (1+x)ⁿ ≈ 1 + nx + n(n−1)x²/2 for |x| < 1. Used in: relativity (expanding the Lorentz factor), uncertainty analysis, AC circuit approximations. Euler's formula: eⁱˣ = cos(x) + i·sin(x). This emerges directly from comparing the Taylor series of eˣ, cos(x), and sin(x). The most beautiful equat
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