Taylor polynomials are used to approximate functions near a value [latex]x=a[/latex]. Maclaurin polynomials are Taylor polynomials at [latex]x=0[/latex].
The nth degree Taylor polynomials for a function [latex]f[/latex] are the partial sums of the Taylor series for [latex]f[/latex].
If a function [latex]f[/latex] has a power series representation at [latex]x=a[/latex], then it is given by its Taylor series at [latex]x=a[/latex].
A Taylor series for [latex]f[/latex] converges to [latex]f[/latex] if and only if [latex]\underset{n\to \infty }{\text{lim}}{R}_{n}\left(x\right)=0[/latex] where [latex]{R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)[/latex].
The Taylor series for ex, [latex]\sin{x}[/latex], and [latex]\cos{x}[/latex] converge to the respective functions for all real x.
Key Equations
Taylor series for the function [latex]f[/latex] at the point [latex]x=a[/latex]
a Taylor polynomial centered at 0; the [latex]n[/latex]th Taylor polynomial for [latex]f[/latex] at 0 is the [latex]n[/latex]th Maclaurin polynomial for [latex]f[/latex]
Maclaurin series
a Taylor series for a function [latex]f[/latex] at [latex]x=0[/latex] is known as a Maclaurin series for [latex]f[/latex]
Taylor polynomials
the [latex]n[/latex]th Taylor polynomial for [latex]f[/latex] at [latex]x=a[/latex] is [latex]{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}[/latex]
Taylor series
a power series at [latex]a[/latex] that converges to a function [latex]f[/latex] on some open interval containing [latex]a[/latex]
Taylor’s theorem with remainder
for a function [latex]f[/latex] and the nth Taylor polynomial for [latex]f[/latex] at [latex]x=a[/latex], the remainder [latex]{R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)[/latex] satisfies [latex]{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}[/latex]
for some [latex]c[/latex] between [latex]x[/latex] and [latex]a[/latex]; if there exists an interval [latex]I[/latex] containing [latex]a[/latex] and a real number [latex]M[/latex] such that [latex]|{f}^{\left(n+1\right)}\left(x\right)|\le M[/latex] for all [latex]x[/latex] in [latex]I[/latex], then [latex]|{R}_{n}\left(x\right)|\le \frac{M}{\left(n+1\right)\text{!}}{|x-a|}^{n+1}[/latex]
