Understanding the nth-Term Test for Divergence

The nth-term test for divergence is an important idea when studying infinite series. It helps us figure out if a series is behaving properly or not.

First, let’s talk about what an infinite series is.

An infinite series is just the total of the numbers in an endless list. You can write it like this:

$S = a_1 + a_2 + a_3 + \ldots$

In this equation, ( a_n ) means the n-th term of the series.

The main question we want to answer is whether the series converges (means it settles at a specific value) or diverges (means it keeps growing without reaching a limit).

What is the nth-Term Test for Divergence?

The nth-term test for divergence says:

If the limit of the series terms does not get closer to zero, then the series diverges.

In simpler terms:

$\lim_{n \to \infty} a_n \neq 0$

If this limit doesn’t exist or doesn’t equal zero, then the series

$S = \sum_{n=1}^{\infty} a_n$

diverges.

It’s important to remember that this test only shows divergence, not convergence.

Let’s Break it Down

1. Limit Condition: The main idea of this test is about what happens to the terms of the series. For a series to converge, the terms need to get closer and closer to zero. If they don’t, the total won’t settle at a final value.

2. Examples:

   • Take the series $S = \sum_{n=1}^{\infty} 1$. In this case, ( a_n = 1 ) for every ( n ). So,

   $\lim_{n \to \infty} a_n = 1 \neq 0$

   This means that the series clearly diverges.

   • Now, let’s look at the series $S = \sum_{n=1}^{\infty} \frac{1}{n}$. Here, we find:

   $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n} = 0$

   Even though the result shows that the terms approach zero (which is necessary for convergence), it doesn’t mean the series converges. In fact, it diverges.

Limitations of the Test

While the nth-term test is useful, it has its limits.

For example, if

$\lim_{n \to \infty} a_n = 0$

it doesn’t mean the series converges. We would need to use other tests to check for convergence, like the ratio test, the comparison test, or the integral test.

Summary

In short, the nth-term test for divergence is a key first step in checking how infinite series behave. It helps us quickly find series that diverge. This test shows us how the parts of a series connect to their total sum in calculus.