Taylor Series

A Taylor series writes a smooth function as an infinite sum of powers of (x - a), with each coefficient chosen so the polynomial matches the function, its slope, curvature, and higher derivatives at the point a. It works because the derivatives at a encode the local behavior of the function, and the factorials scale each term so the nth derivative of the series agrees with f^(n)(a). When the series converges to the function, it gives an exact representation; otherwise, finite truncations still provide very accurate local approximations.

Use Taylor series when you need a local approximation of a complicated function, especially near a point where derivatives are easy to compute. They are essential in asymptotic analysis, numerical methods, error estimation, and physics/engineering approximations near equilibrium or small perturbations.

• Forgetting to evaluate the derivatives at the expansion point a before forming coefficients
• Omitting the factorial n! in the denominator, which changes every term's size
• Using the series outside its interval/radius of convergence and assuming it always equals the function