# 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

1. $${a_{n + 1}} \lt {a_n}$$ for all $$n$$;
2. $$\lim\limits_{n \to \infty } {a_n} = 0.$$

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

$\lim\limits_{n \to \infty } \left| {{a_n}} \right| = \lim\limits_{n \to \infty } \left| {{{\left( { - 1} \right)}^n}\frac{{{{\sin }^2}n}}{n}} \right| = \lim\limits_{n \to \infty } \frac{{{{\sin }^2}n}}{n} = 0,$

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:

$\lim\limits_{n \to \infty } \left| {{a_n}} \right| = \lim\limits_{n \to \infty } \frac{{2n + 1}}{{3n + 2}} = \lim\limits_{n \to \infty } \frac{{\frac{{2n + 1}}{n}}}{{\frac{{3n + 2}}{n}}} = \lim\limits_{n \to \infty } \frac{{2 + \frac{1}{n}}}{{3 + \frac{2}{n}}} = \frac{2}{3} \ne 0.$

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

See more problems on Page 2.