What is the Taylor Series?
The Taylor series expresses any smooth function as an infinite polynomial. Each coefficient is a derivative: the nth term is f⁽ⁿ⁾(a)/n! times (x-a)ⁿ. For well-behaved functions like eˣ, sin(x), and cos(x), the series converges to the exact function value everywhere.
Each extra term extends the approximation further. Adding more terms: sin(x) ≈ x − x³/6 + x⁵/120 − x⁷/5040 + …
The three most important Maclaurin series: eˣ = 1 + x + x²/2! + x³/3! + ⋯ (converges everywhere); sin(x) = x - x³/3! + x⁵/5! - ⋯ (converges everywhere); cos(x) = 1 - x²/2! + x⁴/4! - ⋯ (converges everywhere). Substituting x = iπ in the eˣ series produces Euler's identity.
| f(x) | Series | Radius |
|---|---|---|
| eˣ | 1+x+x²/2!+x³/3!+⋯ | ∞ |
| sin x | x-x³/3!+x⁵/5!-⋯ | ∞ |
| cos x | 1-x²/2!+x⁴/4!-⋯ | ∞ |
| ln(1+x) | x-x²/2+x³/3-⋯ | |x|≤1 |
| 1/(1-x) | 1+x+x²+x³+⋯ | |x|<1 |
Brook Taylor stated the general theorem in 1715; the special case centred at 0 was popularised by Colin Maclaurin in 1742. Every calculator and computer uses Taylor series to evaluate transcendental functions. The error after n terms is bounded by the Lagrange remainder: |f(x) - Pₙ(x)| ≤ max|f⁽ⁿ⁺¹⁾| · |x-a|ⁿ⁺¹ / (n+1)!
cos(x) ≈ 1 − x²/2 + x⁴/24 − x⁶/720 + … Each pair of terms is one more order of accuracy.
A Taylor series represents a smooth function as an infinite polynomial: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... Coefficients are derivatives at the centre point a. Maclaurin series are centred at 0. The three key series converge everywhere: e^x = 1 + x + x^2/2! + ..., sin(x) = x - x^3/3! + x^5/5! - ..., cos(x) = 1 - x^2/2! + x^4/4! - ... Substituting x = i*pi in the e^x series proves Euler's identity. Every calculator uses Taylor series internally to evaluate transcendental functions.
Pi
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