Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Calculus Volume 2

Key Concepts

Calculus Volume 2Key Concepts

Key Concepts

6.1 Power Series and Functions

  • For a power series centered at x=a,x=a, one of the following three properties hold:
    1. The power series converges only at x=a.x=a. In this case, we say that the radius of convergence is R=0.R=0.
    2. The power series converges for all real numbers x. In this case, we say that the radius of convergence is R=.R=.
    3. There is a real number R such that the series converges for |xa|<R|xa|<R and diverges for |xa|>R.|xa|>R. In this case, the radius of convergence is R.
  • If a power series converges on a finite interval, the series may or may not converge at the endpoints.
  • The ratio test may often be used to determine the radius of convergence.
  • The geometric series n=0xn=11xn=0xn=11x for |x|<1|x|<1 allows us to represent certain functions using geometric series.

6.2 Properties of Power Series

  • Given two power series n=0cnxnn=0cnxn and n=0dnxnn=0dnxn that converge to functions f and g on a common interval I, the sum and difference of the two series converge to f±g,f±g, respectively, on I. In addition, for any real number b and integer m0,m0, the series n=0bxmcnxnn=0bxmcnxn converges to bxmf(x)bxmf(x) and the series n=0cn(bxm)nn=0cn(bxm)n converges to f(bxm)f(bxm) whenever bxm is in the interval I.
  • Given two power series that converge on an interval (R,R),(R,R), the Cauchy product of the two power series converges on the interval (R,R).(R,R).
  • Given a power series that converges to a function f on an interval (R,R),(R,R), the series can be differentiated term-by-term and the resulting series converges to ff on (R,R).(R,R). The series can also be integrated term-by-term and the resulting series converges to f(x)dxf(x)dx on (R,R).(R,R).

6.3 Taylor and Maclaurin Series

  • Taylor polynomials are used to approximate functions near a value x=a.x=a. Maclaurin polynomials are Taylor polynomials at x=0.x=0.
  • The nth degree Taylor polynomials for a function ff are the partial sums of the Taylor series for f.f.
  • If a function ff has a power series representation at x=a,x=a, then it is given by its Taylor series at x=a.x=a.
  • A Taylor series for ff converges to ff if and only if limnRn(x)=0limnRn(x)=0 where Rn(x)=f(x)pn(x).Rn(x)=f(x)pn(x).
  • The Taylor series for ex, sinx,sinx, and cosxcosx converge to the respective functions for all real x.

6.4 Working with Taylor Series

  • The binomial series is the Maclaurin series for f(x)=(1+x)r.f(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.
Order a print copy

As an Amazon Associate we earn from qualifying purchases.


This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at
Citation information

© Feb 5, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.