# Calculus

## Infinite Sequences and Series # 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 $$\sum\limits_{n = 1}^\infty {{{\left( { – 1} \right)}^n}\large\frac{{{{\sin }^2}n}}{n}\normalsize}.$$

### 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
${\frac{2}{{3!}} – \frac{{{2^2}}}{{5!}} }+{ \frac{{{2^3}}}{{7!}} – \frac{{{2^4}}}{{9!}} + \ldots }$

### 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.

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Problems 1-2
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Problems 3-7