Differential Equations Workbook For Dummies®

8.4. Answers to Solving Ordinary Differential Equations with Power Series

Here are the answers to the practice questions I provide throughout this chapter. I walk you through each answer so you can see the problems worked out step by step. Enjoy!

1 Does this series converge?

Answer: Yes, if 4 < x < 6.

1. Take a look at the ratio of the (n + 1)th term to the nth term:

   

2. This ratio becomes

   

3. So the ratio is |x − 5|, and the series converges if that ratio is less than 1. In other words, the range in which the series converges is |x − 5| < 1.

4. Therefore, if x is in the range 4 < x < 6, the series converges.

2 State whether this series converges:

Answer: Yes, if 0 < x < 2.

1. Check out the ratio of the (n + 1)th term to the nth term:

   

2. This ratio works out to

   

3. As you can see, the ratio is |x − 1|; the series converges if that ratio is less than 1. So the range in which the series converges absolutely ...