10.3 The nth Term Test for Divergence

The nth-term test for divergence says that if ${\lim_{n\to\infty} a_n \ne 0}$ or if the limit does not exist, then the series ${\sum a_n}$ must diverge. This is a very useful first test because it is quick to apply and can immediately rule out convergence. However, it only works in one direction: if the terms do go to 0, that does not guarantee the series converges. It only means you need another test.

You should think of this test as a basic screening tool. For an infinite series to have any chance of converging, its terms must become arbitrarily small. If they do not even go to 0, then adding them forever cannot possibly settle to a finite sum. But many series have terms going to 0 and still diverge, so this test can prove divergence but cannot prove convergence.

Example 1: Test ${\sum_{n=1}^{\infty} \frac{n}{n+1}}$ The terms satisfy ${\lim_{n\to\infty} \frac{n}{n+1}=1}$ Since the terms do not go to 0, the series diverges.