How Taylor Series Work
A Taylor series rewrites a smooth function as an infinite polynomial built from its derivatives at a single point: f(x) = f(a) + f′(a)(x−a) + f″(a)(x−a)²/2! + … Each term corrects the approximation using higher-order information about how the function curves. Stop after n terms and you get the Taylor polynomial of degree n.
Maclaurin: The Special Case a = 0
Center at zero and the series is called a Maclaurin series. The famous ones: e^x = 1 + x + x²/2! + x³/3! + …, sin(x) = x − x³/3! + x⁵/5! − …, cos(x) = 1 − x²/2! + x⁴/4! − …. These power series are how calculators themselves evaluate transcendental functions internally.
Convergence Matters
A Taylor polynomial approximates well near the center; far away it can diverge wildly. The interval where the full series converges must be checked separately — try our series convergence calculator for that, or the derivative calculator for the underlying derivatives.