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 an = n as n goes to ∞. Since
the series diverges.