Understanding the nth Term Test for Convergence and Divergence in Math

nth term test

The nth term test is a criterion used to determine the convergence or divergence of a series

The nth term test is a criterion used to determine the convergence or divergence of a series. It states that if the nth term (denoted as aₙ) of a given series does not approach zero as n approaches infinity, then the series is divergent.

In simpler terms, if the individual terms of a series do not approach zero, then the series will not have a finite sum and will diverge. On the other hand, if the terms do approach zero, it does not necessarily mean that the series converges, as there are other convergence tests that need to be checked.

Here’s an example to illustrate this:

Consider the series 1 + 1/2 + 1/3 + 1/4 + …

We can see that as n approaches infinity, the terms become smaller and smaller (1/n) but they never reach zero. In this case, the nth term test tells us that the series diverges since the terms do not approach zero.

It is important to note that the nth term test can only determine divergence; it cannot guarantee convergence. To test for convergence, other tests such as the geometric series test, ratio test, or integral test should be employed.

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