If the individual terms of a series (in other words, the terms of the series’ underlying sequence) do not converge to zero, then the series must diverge. This is the *n*th term test for divergence. This is usually a *very* easy test to use.

**The ***n***th Term Test:**

(You probably figured out that with this naked summation symbol, *n* runs from 1 to infinity.)

If you think about it, this is just common sense. When a series converges, the sum of all the terms is honing in on a certain number. The only way this can happen is when the numbers being added at the far “end” of the series are getting infinitesimally small — like in the series:

Imagine, instead, that the terms of a series are converging, say, to 1, like in the series

In that case, when you add up the terms, you keep adding on numbers extremely close to 1 over and over and over forever — and this must add up to infinity. So, in order for a series to converge, the terms of the series must converge to zero. But make sure you understand what this *n*th term test does *not* say.

When the terms of a series converge to zero, that does not guarantee that the series converges. In hifalutin logicianese — the fact that the terms of a series converge to zero is a *necessary* but *not** sufficient* condition for concluding that the series converges to a finite sum.

Because this test is usually very easy to apply, it should be one of the first things you check when trying to determine whether a series converges or diverges. For example, if you’re asked to determine whether

converges or diverges, note that every term of this series is a number greater than 1 being raised to a positive power. This always results in a number greater than 1, and thus, the terms of this series do not converge to zero, and the series must therefore diverge.

The *n*th term test not only works for ordinary positive series, but it also works for series with positive and negative terms.