Question

Write a power series that has the indicated interval of convergence. Explain your reasoning. (a) (-2,2) (b) (-1,1] (c) (-1,0) (d) [-2,6)

Short answer

The power series for each given interval are: (a) \( \sum_{n=0}^{\infty} x^{n} \), (b) \( \sum_{n=0}^{\infty} (-1)^{n}x^{n} \), (c) \( \sum_{n=0}^{\infty} (x+0.5)^{n} \), and (d) \( \sum_{n=0}^{\infty} (-1)^{n}(x-2)^{n} \)

Step-by-step solution

Solve for (a) (-2,2)

First, note that the interval (-2,2) has a length of 4, and hence the radius of convergence is 2. The series will be centered at 0. Considering that the convergence does not include the end points, we get the power series \( \sum_{n=0}^{\infty} x^{n} \), because it converges for |-x|<2 which implies -2<x<2.

Solve for (b) (-1,1]

For the interval (-1,1], the radius of convergence is 1 and the series is centered at 0. The convergence includes the endpoint 1, not -1. Thus, we can get the power series with \( \sum_{n=0}^{\infty} (-1)^{n}x^{n} \) because it converges for |-x|<=1 which implies -1<x<=1.

Solve for (c) (-1,0)

For the interval (-1,0), the radius of convergence is 0.5 and the series is centered at -0.5. The convergence does not include the end points. Hence, the power series with \( \sum_{n=0}^{\infty} (x+0.5)^{n} \) is the solution, because |-x+0.5|<0.5 which implies -1<x<0.

Solve for (d) [-2,6)

The interval [-2,6) has a length of 8, hence the radius of convergence is 4 and centered at 2. The convergence includes the endpoint -2, not 6. We can use the power series \( \sum_{n=0}^{\infty} (-1)^{n}(x-2)^{n} \), as it covers the interval -2<=x<6.