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 limn→∞Rn(x)=0 where Rn(x)=f(x)−pn(x).
  • The Taylor series for ex, sinx, and cosx converge to the respective functions for all real x.

Key Equations

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

    ∞∑n=0f(n)(a)n!(x−a)n=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n+⋯

Glossary

Maclaurin polynomial
a Taylor polynomial centered at 0; the nth Taylor polynomial for f at 0 is the nth 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 nth Taylor polynomial for f at x=a is pn(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)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 Rn(x)=f(x)−pn(x) satisfies Rn(x)=f(n+1)(c)(n+1)!(x−a)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(n+1)(x)|≤M for all x in I, then |Rn(x)|≤M(n+1)!|x−a|n+1