Summary of Working with Taylor Series

Essential Concepts

  • The binomial series is the Maclaurin series for f(x)=(1+x)rf(x)=(1+x)r. It converges for |x|<1|x|<1.
  • Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.
  • Power series can be used to solve differential equations.
  • Taylor series can be used to help approximate integrals that cannot be evaluated by other means.

Glossary

binomial series
the Maclaurin series for f(x)=(1+x)rf(x)=(1+x)r; it is given by

(1+x)r=∞∑n=0(rn)xn=1+rx+r(r−1)2!x2+⋯+r(r−1)⋯(r−n+1)n!xn+⋯(1+x)r=∞∑n=0(rn)xn=1+rx+r(r−1)2!x2+⋯+r(r−1)⋯(r−n+1)n!xn+⋯ for |x|<1|x|<1
nonelementary integral
an integral for which the antiderivative of the integrand cannot be expressed as an elementary function