Taylor Series
Definition
Approximating curvy functions with simple polynomials — how physicists tame sines, cosines, and exponentials.
Technical definition
f(x) = Σ f⁽ⁿ⁾(a)(x−a)ⁿ/n!. The small-angle sin θ ≈ θ and e^x ≈ 1 + x are first-order Taylor truncations.
Example
The pendulum period formula uses sin θ ≈ θ — the small-angle approximation is just the Taylor series cut after its first term.
Common confusion
Taylor series don't always converge to the function everywhere — some diverge beyond a radius. Truncation error is real; always check the remainder.