The Lagrange Error Bound

A Taylor polynomial gives an estimate. The Lagrange error bound says how far off that estimate can be, using only a bound on one more derivative.

The right side looks like the next term of the Taylor polynomial, with $f^{(n+1)}(a)$ replaced by $M$. The error itself equals that next-term form evaluated at some unknown $z$ between $a$ and $x$. Since $z$ is unknown, $M$ has to cover every possible $z$, so take the largest value of $|f^{(n+1)}|$ on the interval, or any number above it.

Here the bound is close to the actual error. It can be much larger for other functions, and that's expected, since it has to hold whichever $z$ the error comes from.

Sometimes the terms of a Taylor polynomial, evaluated at a particular $x$, are the first terms of a convergent alternating series whose sum is $f(x)$. If that series passes the alternating series test, the error is at most the first term left out. This is often easier than finding $M$.

A Maclaurin series continues a Maclaurin polynomial forever. Its coefficients are $\frac{g^{(n)}(0)}{n!}$, so its first $n + 1$ terms are $P_n$. In the next example the series and where it converges are given.