Local, and only local
The readout gives how far the approximation stays within 0.1 of the truth, and this is the number worth watching.
More terms extend the range. They do not make it infinite, and the extension gets slower. The polynomial agrees perfectly at the expansion point and gets worse with distance, always — the whole construction is built from information at a single point, so it cannot know anything about elsewhere.
Move the expansion point and the region of agreement moves with it. That is the right mental model: a Taylor series is a statement about a neighbourhood.
Where it earns its keep
Second-order optimisation. Newton's method is exactly the two-term expansion of the loss: approximate it as a quadratic near the current point, jump to that quadratic's minimum, repeat. Everything in [the Hessian module](jacobian_and_hessian.html) is this idea.
Small-angle approximations. sin(x) ~ x for small x is the one-term expansion, and it is why pendulum equations are solvable at all.
Backpropagation's justification. The chain rule is exact, but the argument that a small weight change produces a proportional loss change is a first-order expansion.
Numerical methods. Finite differences, Runge-Kutta integrators and most error bounds are derived by expanding and discarding.
exp, log, sin in a standard library. Not looked up in a table; evaluated from a truncated series with a range reduction in front of it.
Where it fails
Switch the function control to 1 / (1 + x squared).
This function is smooth everywhere on the real line — no kinks, no asymptotes, differentiable as many times as you like. Yet expand it about 0 and add terms, and beyond about x = 1 the approximation does not merely fail to improve; it gets worse with every term added.
The reason is invisible from the real line. Viewed over the complex numbers the function has poles at *i* and *−i*, distance 1 from the origin, and the radius of convergence is the distance to the nearest singularity wherever it sits. That radius is 1, and no number of real terms escapes it.
This is the Runge phenomenon, and its lesson is practical: smooth on the reals does not mean a Taylor series converges everywhere. Adding terms is not always progress.
Where it goes wrong
Extrapolating far from the expansion point. The error grows with distance, and past the radius of convergence it grows without bound.
Assuming more terms is always better. Not past the radius, and not numerically — high-order terms involve large factorials and cancelling quantities, which loses precision in floating point.
Expanding about the wrong point. Expand about where you will evaluate. A series about 0 is the wrong tool for estimating at x = 5.
Forgetting smoothness is required. The function needs derivatives of every order at the expansion point. |x| has none at 0, so there is no series there.