Taylor Series Explorer

A Taylor polynomial is built from derivatives at one point, so it captures the function's local geometry there. Farther away, the same coefficients can stop describing the function well: the error depends on smoothness, degree, and radius of convergence.

This tool rebuilds a curvy function using nothing but additions, multiplications, and powers — the arithmetic a processor can actually do. The degree-0 term matches the function's height at the center, degree 1 matches its slope, degree 2 its bend, and each higher term pins down one more derivative. Raise the degree n and watch the dashed polynomial peel closer to the solid curve near the center point a.

You might expect that an infinitely smooth function is always captured by a high-enough-degree Taylor polynomial — but it isn't guaranteed. The classic counterexample is f(x)=e−1/x² (with f(0)=0): every derivative at 0 equals zero, so its Taylor polynomial around 0 is just 0 at every degree, yet the function is positive everywhere else. Adding terms never helps. Being 'smooth' (all derivatives exist) is weaker than being 'analytic' (actually equal to your own series).