A Taylor series expands any smooth function as an infinite sum of polynomial terms centred at a point. The Maclaurin series (centred at zero) for sin(x) is x−x³/3!+x&sup5;/5!−… Each additional term widens the region where the polynomial faithfully tracks the true curve — the orange line chases the blue one outward as N grows.
Choose a function (sin, cos or e⊃x), then drag the Terms (N) slider to add polynomial terms one by one. Press Animate to watch the approximation step through N=0 to your chosen maximum, revealing how each term improves accuracy. The Max error stat shows the largest deviation across the visible window.
Brook Taylor published the series in 1715, but James Gregory had used special cases decades earlier. The exponential e⊃x has the simplest Taylor series — every coefficient is 1/k! — because e⊃x is its own derivative. Both sin and cos converge globally but alternate in sign, causing the oscillatory overshoot visible at small N far from the origin.