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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 |x−a|<R|x−a|<R and diverges for |x−a|>R.|x−a|>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=0∞xn=11−x∑n=0∞xn=11−x 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=0∞cnxn∑n=0∞cnxn and ∑n=0∞dnxn∑n=0∞dnxn 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 m≥0,m≥0, the series ∑n=0∞bxmcnxn∑n=0∞bxmcnxn converges to bxmf(x)bxmf(x) and the series ∑n=0∞cn(bxm)n∑n=0∞cn(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 f′f′ on (−R,R).(−R,R). The series can also be integrated term-by-term and the resulting series converges to ∫f(x)dx∫f(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 limn→∞Rn(x)=0limn→∞Rn(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.
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