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 a problem to see the solution.

### Example 1

Use the alternating series test to determine the convergence of the series \(\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}\large\frac{{{{\sin }^2}n}}{n}\normalsize}.\)### Example 2

Determine whether the series \(\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}\large\frac{{2n + 1}}{{3n + 2}}\normalsize} \) is absolutely convergent, conditionally convergent, or divergent.### Example 3

Determine whether \(\sum\limits_{n = 1}^\infty {\large\frac{{{{\left( { – 1} \right)}^{n + 1}}}}{{n!}}\normalsize} \) is absolutely convergent, conditionally convergent, or divergent.### Example 4

Determine whether the alternating series \(\sum\limits_{n = 2}^\infty {\large\frac{{{{\left( { – 1} \right)}^{n + 1}}\sqrt n }}{{\ln n}}\normalsize} \) is absolutely convergent, conditionally convergent, or divergent.### Example 5

Determine the \(n\)th term and test for convergence the series### Example 6

Investigate whether the series \(\sum\limits_{n = 1}^\infty {\large\frac{{{{\left( { – 1} \right)}^{n + 1}}}}{{5n – 1}}\normalsize} \) is absolutely convergent, conditionally convergent, or divergent.### Example 7

Determine whether the alternating series \(\sum\limits_{n = 1}^\infty {\large\frac{{{{\left( { – 1} \right)}^n}}}{{\sqrt {n\left( {n + 1} \right)} }}\normalsize} \) is absolutely convergent, conditionally convergent, or divergent.### Example 1.

Use the alternating series test to determine the convergence of the series \(\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}\large\frac{{{{\sin }^2}n}}{n}\normalsize}.\)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 \(\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}\large\frac{{2n + 1}}{{3n + 2}}\normalsize} \) 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.}

\]

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