The terms look roughly like or , so we'll guess that this series diverges. To be convincing, we need to find a series with smaller terms whose sum diverges. Since *n*^{2} – 81 < *n*^{2}
it follows that Since diverges and has smaller terms than the series we were given, the series we were given diverges. In the previous example, we didn't have the relationship for all *n*. The terms are negative for 1 ≤ *n* < 9, and not even defined for *n* = 9. However, the relationship does hold for *n *≥ 10, and that's good enough. There's one down side to the comparison test. When using it, sometimes we have to use other tests also to show the convergence or divergence of the series we're comparing to. Because of this, the comparison test is meant to be a last resort. |