truncation error
Truncation error is the error you make by cutting an infinite or continuous mathematical process down to a finite, computable one — the difference between the exact mathematical operation and the finite recipe you replaced it with. It is the error of the method itself, present even on a perfect machine with no rounding at all.
It appears whenever you stop an infinite sum after a few terms, or replace a limit by a finite-step approximation. Taylor's theorem is the usual lens: approximate f(x+h) by f(x) + h f'(x) and you have dropped the tail h^2 f''(x)/2 + ..., so the truncation error is O(h^2). A forward difference (f(x+h) - f(x))/h estimates f'(x) with error O(h); a central difference (f(x+h) - f(x-h))/(2h) does better, with error O(h^2). The exponent p in O(h^p) is the order of accuracy: the higher the p, the faster the truncation error vanishes as the step h shrinks — halving h cuts an O(h^2) error by four.
Truncation error is the one you control by design: take more terms, use a smaller step, or pick a higher-order method, and it falls predictably. That makes it the friendliest of the errors — but it does not live alone. Shrinking h forever does not keep helping, because round-off error grows as h gets tiny; the two trade off, and the smart step size balances them rather than minimizing truncation error blindly.
Estimating f'(1) for f(x) = e^x by (f(1+h) - f(1))/h: with h = 0.1 the error is about 0.14 (order h), with h = 0.01 about 0.014 — tenfold smaller for tenfold smaller h, exactly as O(h) predicts.
The error of cutting a limit short, falling at the rate the order of accuracy promises.
Do not push h to zero to kill truncation error: below an optimal h, round-off from finite arithmetic takes over and the total error climbs again. The best step balances truncation against round-off.