Calculus
Taylor Series
A Taylor series rebuilds a function out of polynomial pieces. Add terms one at a time and watch the approximation cling to the curve near the centre, then break away once you leave the radius of convergence.
Terms3
Error at x = 1.5—
Teal is the true function, indigo is the polynomial. The shaded band, when shown, is the radius of convergence around x = 0.
What to observe
- Add terms and watch the polynomial match the curve near x = 0 first, spreading outward as the degree grows. Each term fixes one more derivative at the centre.
- For sin, cos, eˣ the series converges everywhere: the error at x = 1.5 keeps shrinking toward zero.
- For ln(1+x) and 1/(1−x) the radius of convergence is 1. Outside the shaded band the polynomial diverges no matter how many terms you add, the error at x = 1.5 blows up.
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