# Alternating Series

A series in which successive terms have opposite signs is called an alternating series.

## The Alternating Series Test (Leibniz's Theorem)

This test is the sufficient convergence test. It's also known as the Leibniz's Theorem for alternating series.

Let \(\left\{ {{a_n}} \right\}\) be a sequence of positive numbers such that

- \({a_{n + 1}} \lt {a_n}\) for all \(n\);
- \(\lim\limits_{n \to \infty } {a_n} = 0.\)

Then the alternating series \(\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}{a_n}} \) and \(\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^{n - 1}}{a_n}} \) both converge.

## Absolute and Conditional Convergence

A series \(\sum\limits_{n = 1}^\infty {{a_n}}\) is absolutely convergent, if the series \(\sum\limits_{n = 1}^\infty {\left| {{a_n}} \right|} \) is convergent.

If the series \(\sum\limits_{n = 1}^\infty {{a_n}}\) is absolutely convergent then it is (just) convergent. The converse of this statement is false.

A series \(\sum\limits_{n = 1}^\infty {{a_n}}\) is called conditionally convergent, if the series is convergent but is not absolutely convergent.

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Use the alternating series test to determine the convergence of the series \[\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n} \frac{{{{\sin }^2}n}}{n}}.\]

### Example 2

Determine whether the series \[\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}\frac{{2n + 1}}{{3n + 2}}} \] is absolutely convergent, conditionally convergent, or divergent.

### Example 1.

Use the alternating series test to determine the convergence of the series \[\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n} \frac{{{{\sin }^2}n}}{n}}.\]

Solution.

By the alternating series test we find that

since \({\sin ^2}n \le 1.\) Hence, the given series converges.

### Example 2.

Determine whether the series \[\sum\limits_{n = 1}^\infty {{{\left( { - 1} \right)}^n}\frac{{2n + 1}}{{3n + 2}}} \] is absolutely convergent, conditionally convergent, or divergent.

Solution.

We try to apply the alternating series test here:

The \(n\)th term does not approach \(0\) as \(n \to \infty.\) Therefore the given series diverges.