ISPA - Power Series and Convergence

Power Series and Convergence

Power Series and Convergence

A power series is an infinite series of the form $\sum_{n=0}^\infty c_nx^n=c_0+c_1x+c_2x^2+c_3x^3+...$ where x is a variable and c_n's are constants (coefficients). The power series $\sum_{n=0}^\infty c_nx^n=c_0+c_1x+c_2x^2+c_3x^3+...$ is said to be centered at 0, although that is not a requirement to be a power series. A power series may converge for some values of x and diverge for other x values. The sum of a power series is a function f(x) = c₀ + c₁x + c₂x² + c₃x³ + ... + c_nxⁿ + ... whose domain is the set of all x for which the series converges. Finding the domain of a power series function for which it converges is of particular interest. Note that f(x) would be called a polynomial if it did not contain an infinite number of terms.

Since power series involve powers of x, the Ratio Test is particularly effective in deciding the values of x for which a power series converges. The presentation below provides three examples of using the Ratio Test to determine the values of x for which a power series centered at 0 converges.

In general, a series of the form $\sum_{n=0}^\infty c_n(x-a)^n=c_0+c_1(x-a)+c_2(x-a)^2+c_3(x-a)^3+...$ is a power series centered at a or a power series about a.

Radius and Interval of Convergence

As seen in the three examples above, a power series always behaves in one of three ways:

1. The series converges at x = a and diverges elsewhere.

2. The series converges absolutely for all x.

3. There is a positive number R such that the series converges if |x - a|< R and diverges if |x - a|> R. The series may or may not converge at either of the endpoints x = a - R and x = a + R.

The number R above is called the radius of convergence, which is the radius of the interval around a given point a such that a given power series $\sum_{n=0}^\infty c_n(x-a)^n$ converges absolutely at all points inside the interval. The interval of convergence is the set of all values of x for which the power series $\sum_{n=0}^\infty c_n(x-a)^n$ centered at a converges. In the case where the series converges at x = a, the interval of convergence consists of only a single point a and the radius of convergence is R = 0. When the series converges absolutely for all x, the interval of convergence is $\left(-\infty,\:\infty\right)$ and the radius of convergence is . For the geometric series in Example 3 above, $\sum_{n=0}^\infty x^n=1+x+x^2+...$ , the radius of convergence is 1 and the interval of convergence is (-1, 1).

Examples of using the Ratio Test to determine the radius and interval of convergence are the focus of the next video.

Endpoint Convergence

Recall that the radius of convergence, R, defines the boundaries of the interval of convergence and |x - a| < R can be rewritten as a - R < x < a + R. When x is an endpoint of the interval, x = a ± R, the series may converge at one or both endpoints or it may diverge at both endpoints. There are four possibilities for the interval of convergence: (a - R, a + R), (a - R, a + R], [a - R, a + R), or [a - R, a + R]. Since the Ratio Test always fails when x is an endpoint of the interval of convergence, the endpoint must be checked with another test.

View the next two presentations on power series and radius and interval of convergence.

Two examples of finding the radius and interval of convergence of power series are presented below.

Use the interactive applet to explore power series and interval of convergence by CLICKING HERE. Links to an external site.

Power Series and Convergence Practice

Power Series and Convergence: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of power series and convergence.

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