Summary of Taylor and Maclaurin Series

Essential Concepts

• Taylor polynomials are used to approximate functions near a value $x=a$. Maclaurin polynomials are Taylor polynomials at $x=0$.
• The nth degree Taylor polynomials for a function $f$ are the partial sums of the Taylor series for $f$.
• If a function $f$ has a power series representation at $x=a$, then it is given by its Taylor series at $x=a$.
• A Taylor series for $f$ converges to $f$ if and only if $\underset{n\to \infty }{\text{lim}}{R}_{n}\left(x\right)=0$ where ${R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)$.
• The Taylor series for ex, $\sin{x}$, and $\cos{x}$ converge to the respective functions for all real x.

Key Equations

• Taylor series for the function $f$ at the point $x=a$

$\displaystyle\sum _{n=0}^{\infty }\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}=f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)+\frac{f^{\prime\prime}\left(a\right)}{2\text{!}}{\left(x-a\right)}^{2}+\cdots +\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}+\cdots$

Glossary

Maclaurin polynomial
a Taylor polynomial centered at 0; the $n$th Taylor polynomial for $f$ at 0 is the $n$th Maclaurin polynomial for $f$
Maclaurin series
a Taylor series for a function $f$ at $x=0$ is known as a Maclaurin series for $f$
Taylor polynomials
the $n$th Taylor polynomial for $f$ at $x=a$ is ${p}_{n}\left(x\right)=f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)+\frac{f^{\prime\prime}\left(a\right)}{2\text{!}}{\left(x-a\right)}^{2}+\cdots +\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}$
Taylor series
a power series at $a$ that converges to a function $f$ on some open interval containing $a$
Taylor’s theorem with remainder
for a function $f$ and the nth Taylor polynomial for $f$ at $x=a$, the remainder ${R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)$ satisfies ${R}_{n}\left(x\right)=\frac{{f}^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)\text{!}}{\left(x-a\right)}^{n+1}$

for some $c$ between $x$ and $a$; if there exists an interval $I$ containing $a$ and a real number $M$ such that $|{f}^{\left(n+1\right)}\left(x\right)|\le M$ for all $x$ in $I$, then $|{R}_{n}\left(x\right)|\le \frac{M}{\left(n+1\right)\text{!}}{|x-a|}^{n+1}$