# Series

# The Divergence Test

Since an uncontrolled grilled cheese spill is a hazardous materials catastrophe we'd like to avoid, we want a test that will tell us when *not* to open the Pandora's box.

We know that if a series converges, its terms must approach zero. Rephrasing this in the contrapositive, *if the terms of a series don't approach zero, the series diverges.* This statement lets us look at a series and, if the terms don't approach zero, conclude that the series diverges.

**Be Careful:** We can't use this statement to conclude that a series converges. We can only use it to evaluate if a series diverges. That's why we call it the **Divergence Test**. If the terms do approach zero, there's hope that the series might converge, but we would need to use other tools to really draw that conclusion.

### Sample Problem

Does the series

converge or diverge?

Answer.

Look at the limit of the terms *a _{n}* =

*n*as

*n*goes to ∞. Since

the series diverges.