The Divergence Test

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Now that we can find the sum of an infinite series when it converges, we’d like to be able to tell if a given series really does converge. Often it’s not easy to tell. But there’s one great test that can sometimes tell us when a series doesn’t converge. It’s called

The Divergence Test


Let

    \[ \sum_{n=1}^{\infty} a_n \]

be a series. Then if \lim_{n\rightarrow\infty} a_n\neq 0 or doesn’t exist, then the series diverges.

Note: the divergence test can only tell you that a series diverges. It can never tell you that a given sequence converges. That’s why it’s called the “divergence test.” This is super duper important. Don’t make the mistake of writing something like “this sequence converges because \lim_{n\rightarrow\infty} a_n = 0” on your test. This is one of the most common mistakes students make with series.

Let’s look at a couple of examples. Consider the sequence

    \[ \sum_{n=1}^{\infty} \left(1-\frac{1}{n}\right). \]

Let’s do the divergence test. We must compute: \lim_{n\rightarrow\infty}\left(1-\frac{1}{n}\right) = 1\neq 0, so by the divergence test the series diverges. Why does this make sense? Consider large values of n, so that 1/n is very small. Then the series looks almost like this

    \[ \ldots + 1 + 1 + 1 + 1 + \dots \]

I say “almost” because those numbers a just a little smaller than 1. However, it’s pretty clear that such a series won’t converge: the partial sums are getting larger and larger.

Problem: Consider the series

    \[ \sum_{n=1}^{\infty} \frac{1}{n}. \]

What does the divergence test say about this series?

Answer: Nothing! We can compute the limit: \lim_{n\rightarrow\infty}\frac{1}{n} = 0, but this doesn’t tell us that the series converges. It also doesn’t tell us that the series diverges. It doesn’t tell us anything at all! Mystery!