Introduction

Welcome to this guide on the $n$th term test for divergence. This article will guide you through how to perform a $n$th term test to determine a convergence of a series.

nth Term Test

The $n$th term test goes as follows:

For an infinite series $\sum _{n=1}^{\infty }{a}_n$ , if $\underset{x\to \infty }{\lim }{a}_n\ne 0$ or the limit does not exist, then $\sum _{n=1}^{\infty }{a}_n$ diverges.

This theorem is the $n$th term test, it is important to point out that if the limit equals $0$, this test cannot draw any conclusion.

If $\underset{x\to \infty }{\lim }{a}_n=0$, then $\sum _{n=1}^{\infty }{a}_n$ doesn’t have to converge.

Note that zero/one-based indexing here does not matter, since shifting the starting index only affects one finite term. However, as the series are infinite, that starting finite term will not impact whether the series ultimately diverges or converges.

Using the nth Term Test

Example 1:

$\sum _{n=1}^{\infty }\left(\frac{n}{n+100}\right)$

By the $n$th Term Test, we know that:

$\underset{n\to \infty }{\lim }\frac{n}{n+100}=\underset{n\to \infty }{\lim }\frac{1}{1}=1$

The limit does not equal to $0$, therefore this series diverges.

Example 2:

$\sum _{n=1}^{\infty }{\left(\frac{1}{n}\right)}^2$

Using the $n$th term test to determine whether or not this series diverges, we will examine this limit:

$\underset{x\to \infty }{\lim }{\left(\frac{1}{n}\right)}^2=0$

The limit equals to $0$, which means the $n$th term test cannot draw any conclusion. This series does converge, which we will prove through later tests.

Practice

Determine if the following series diverge using the $n$th Term Test