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.
the true function fthe running sum Psingularity
Drag across the graph or the lanes to read the gap at another x. Each lane is one term on its own (squashed to fit), bright where it is bigger than 0.05.
Gap at x = 1.50—
—
Function
degree of P—
f(x)—
P(x)—
radius of convergence—
What to observe
- Watch each new term drop from its lane into the sum. A term is tiny near 0 and only matters further out, so every one pushes the stretch where P hugs f a little wider.
- Each lane's coefficient is f⁽ᵏ⁾(0)/k!: for eˣ every derivative at 0 is 1, so the terms are xᵏ/k!, and each one makes P match one more derivative. With no singularity anywhere the disc fills the plane, and x = 2.5 converges too.
- ln(1+x) is perfectly fine at x = 1.5, yet the gap there grows with every term. The series cannot reach past a point where the function blows up: z = −1 caps the disc at radius 1, on both sides of 0.
- 1/(1+x²) is smooth on the whole real line, but its series still dies at |x| = 1. The culprits are the poles at ±i, off to the side, at distance 1 from the centre.
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