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A Taylor series expresses any smooth function as an infinite sum of polynomial terms, allowing complex functions to be approximated by simpler polynomials. Centered at x = a: f(x) = Σₙ₌₀∞ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ, where f⁽ⁿ⁾(a) is the n-th derivative of f evaluated at a.
A Maclaurin series is a Taylor series centered at a = 0 — the most common case. Key Maclaurin series: eˣ = 1 + x + x²/2! + x³/3! + …, sin(x) = x − x³/3! + x⁵/5! − …, cos(x) = 1 − x²/2! + x⁴/4! − …
Taylor polynomials are finite truncations: Tₙ(x) = Σₖ₌₀ⁿ f⁽ᵏ⁾(a)/k! · (x−a)ᵏ. The error (remainder) is bounded by Taylor's theorem: |Rₙ(x)| ≤ M|x−a|ⁿ⁺¹/(n+1)!, where M bounds |f⁽ⁿ⁺¹⁾|.
Applications: calculators approximate trig and log functions using Taylor polynomials; physics uses Taylor expansions to simplify complex models near equilibrium; machine learning uses them in optimization theory.
f(x) = Σₙ₌₀∞ [f⁽ⁿ⁾(a) / n!] · (x − a)ⁿ
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.