Error Bounds for Taylor Polynomials Cheat Sheet

Key Facts

• The degree $n$ Taylor polynomial for $f$ centered at $a$ is $T_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$.
• The exact error, or remainder, is $R_n(x)=f(x)-T_n(x)$.
• Taylor's theorem with Lagrange remainder gives $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$.
• If $|f^{(n+1)}(t)|\le M$ for all $t$ between $a$ and $x$, then $|R_n(x)|\le \frac{M|x-a|^{n+1}}{(n+1)!}$.
• An approximation interval can be written as $f(x)\in \left[T_n(x)-B,\,T_n(x)+B\right]$ when $|R_n(x)|\le B$.
• For an alternating series with decreasing terms approaching $0$, the error after $n$ terms satisfies $|R_n|\le$ the first omitted term.
• For Maclaurin polynomials, the center is $a=0$, so the error bound becomes $|R_n(x)|\le \frac{M|x|^{n+1}}{(n+1)!}$.
• To guarantee accuracy within a tolerance $\varepsilon$, choose $n$ so that $\frac{M|x-a|^{n+1}}{(n+1)!}\le \varepsilon$.

Vocabulary

Taylor Polynomial

A polynomial $T_n(x)$ built from the derivatives of a function at a center $a$ to approximate the function near $a$.

Maclaurin Polynomial

A Taylor polynomial centered at $a=0$.

Remainder

The error term $R_n(x)=f(x)-T_n(x)$ that measures the difference between the function and its Taylor polynomial.

Lagrange Error Bound

An inequality using a bound on $|f^{(n+1)}(x)|$ to limit the possible size of the Taylor polynomial error.

Tolerance

A maximum allowed error, often written as $\varepsilon$, that an approximation must satisfy.

First Omitted Term

In an alternating series estimate, the next term not included in the partial sum, which bounds the absolute error when the conditions hold.