Calculus II For Dummies, 2nd Edition

Chapter 12

Where Is This Going? Testing for Convergence and Divergence

In This Chapter

Understanding convergence and divergence

Using the nth-term test to prove that a series diverges

Applying the versatile integral test, ratio test, and root test

Distinguishing absolute convergence and conditional convergence

Testing for convergence and divergence is The Main Event in your Calculus II study of series. Recall from Chapter 11 that when a series converges, it can be evaluated as a real number. However, when a series diverges, it can’t be evaluated as a real number, because it either explodes to positive or negative infinity or fails to settle in on a single value.

In Chapter 11, I give you two tests for determining whether specific types of series (geometric series and p-series) are convergent or divergent. In this chapter, I give you seven more tests that apply to a much wider range of series.

The first of these is the nth-term test, which is sort of a no-brainer. With this test under your belt, I move on to two comparison tests: the direct comparison test and the limit comparison ...