Taylor polynomials of sin x

This animation overlays the sine curve with its Taylor polynomials of degree 1, 3, 5, 7, 9, and 11, drawn one after another. Each added term visibly widens the interval where the polynomial closely tracks the true sine curve, while the fit still fails far from the center. It helps students see convergence of a power series as a local-to-global process, useful for calculus courses covering series expansions and approximation.

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The prompt that made it

The Taylor polynomials of sin x of degree 1, 3, 5, 7, 9 and 11 drawn over the sine curve: each added term widens the interval where the polynomial matches the function.

Make your own version

Make the next one in this series

• Show the error term between sin x and its Taylor polynomial shrinking as degree increases
• Compare Taylor polynomials of cos x and sin x approximating together on the same axes
• Animate the radius of convergence concept using e^x and 1/(1-x) Taylor series

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