Divergence Test

Series Divergence Tests

Here you will see a test that is only good to tell if a series diverges. Consider the series

\[\sum_{n=1}^{\infty} a_n,\]

and call the partial sums for this series \(s_n\). Sometimes you can look at the limit of the sequence \({a_n}\) to tell if the series diverges. This is called the \(n^{th}\) term test for divergence.

\(n^{th}\) term test for divergence.

If

\[\lim\limits_{n\to \infty}a_n\]

does not exist, or if it does exist but is not equal to zero, then the series

\[\sum_{n=1}^{\infty}a_n\]

diverges.

Proof of the \(n^{th}\) Term Test for Divergence

Let's take a look at the \(n^{th}\) term test for divergence is true. Sometimes in math, you prove a statement like "if A is true then B is true", and sometimes it is easier to prove the contrapositive, which is "if B is false then A is false".

For the \(n^{th}\) term test for divergence, it is easier to show the contrapositive.

So what is the contrapositive for the \(n^{th}\) term test for divergence?

The statement B is "the series diverges", and saying that "B is false" is the same as saying "the series converges".

The statement A is "the limit of the sequence either doesn't exist or does exist and isn't zero", and "A is false" is the same as saying "the limit of the sequence is zero". That means we will look at the proof of:

If \(\sum\limits_{n=1}^{\infty}a_n\) converges then

\[\lim\limits_{n\to \infty}a_n=0.\]

To do this, you will need to look at the partial sums for the series. The sequence of partial sums is defined by \[s_n=\sum_{k=1}^{n}a_k.\]

The previous term in the sequence of partial sum would be \[s_{n-1}=\sum_{k=1}^{n-1}a_k\]

Subtracting them gives,

\[\begin{align} s_{n}-s_{n-1}&=\sum_{k=1}^{n} a_{k}-\sum_{k=1}^{n-1}a_{k}\\&=(a_{1}+a_{2}+\cdots+a_{n-1}+a_{n})-(a_{1}+a_{2}+\cdots+a_{n-1})\\&=a_{n}.\end{align}\]

You already know that the series converges, which implies that the sequence of partial sums converges as well, or in other words

\[\lim\limits_{n\to\infty}s_n=L\]

for some real number \(L\). Now taking the limit of the subtraction of partial sums,

\[\begin{align}\lim\limits_{n\to\infty}[s_n-s_{n-1}]&=\lim\limits_{n\to \infty}s_n-\lim\limits_{n\to\infty}s_{n-1}\\&=L-L\\&=0.\end{align}\]

But you also know that

\[\lim\limits_{n\to\infty}[s_{n}-s_{n-1}]=\lim\limits_{n\to\infty} a_n,\]

which means that

\[\lim\limits_{n\to\infty}a_n=0.\]

Examples Using the \(n^{th}\)Term Test for Divergence

Let's look at some examples of how to properly use the \(n^{th}\) term test for divergence.

Let's take a look at another example.

Integral Divergence Test

As mentioned already, the Integral Test has a part that talks about divergence. So for more information on divergence when using the Integral Test, see Integral Test.