6.1 Power Series and Functions
- For a power series centered at one of the following three properties hold:
- The power series converges only at In this case, we say that the radius of convergence is
- The power series converges for all real numbers x. In this case, we say that the radius of convergence is
- There is a real number R such that the series converges for and diverges for 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 for allows us to represent certain functions using geometric series.
6.2 Properties of Power Series
- Given two power series and that converge to functions f and g on a common interval I, the sum and difference of the two series converge to respectively, on I. In addition, for any real number b and integer the series converges to and the series converges to whenever bxm is in the interval I.
- Given two power series that converge on an interval the Cauchy product of the two power series converges on the interval
- Given a power series that converges to a function f on an interval the series can be differentiated term-by-term and the resulting series converges to on The series can also be integrated term-by-term and the resulting series converges to on
6.3 Taylor and Maclaurin Series
- Taylor polynomials are used to approximate functions near a value Maclaurin polynomials are Taylor polynomials at
- The nth degree Taylor polynomials for a function are the partial sums of the Taylor series for
- If a function has a power series representation at then it is given by its Taylor series at
- A Taylor series for converges to if and only if where
- The Taylor series for ex, and converge to the respective functions for all real x.
6.4 Working with Taylor Series
- The binomial series is the Maclaurin series for It converges for
- 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.