Stand at one point on a curve and measure its value, its slope, its bend, its change of bend — all its derivatives. Astonishingly, that is enough to reconstruct the whole function as a polynomial. Each term you add matches one more derivative and the polynomial hugs the curve a little further out. Add terms and watch it snap onto the shape.
Drag on the plot to move the center a. a = 0 is a Maclaurin series.
f(x) — the true functionTₙ(x) — Taylor polynomialThe Taylor series is f(x) = Σ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ: the nth term is set by the nth derivative at the center a, and dividing by n! is what keeps the higher terms from exploding. Near a even one or two terms are excellent — that's why physics linearizes (sin θ ≈ θ) and why calculators evaluate eˣ and sin x from just a handful of terms. But the match has a range: 1/(1−x) and ln(1+x) only converge for |x−a| within a radius of convergence, and outside it the polynomial diverges no matter how many terms you add — drag the degree up on those two and watch the tails fly off. This single idea — approximate the complicated by the polynomial that matches it locally — underlies numerical methods, series solutions of differential equations, and half of applied analysis.