Alternating Series Test |

**Alternating Series**

Some series have alternating positive and negative terms. These are called **alternating series**. They are identified by the term ( – 1)* ^{n} *or ( – 1)

**The Alternating Series Test**

If the terms are **decreasing** -- so that for every ** k**, we need only show that the limit of the

**The Alternating Series Test** (AST)

The series , with *a _{n}* > 0, converges if for every

Note: this test applies **only to an alternating series**. Before we apply the test, we must identify the series as alternating and we must state that the terms of the series are decreasing. Then we apply the test to the **positive term series** omitting the ( – 1) term.

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**Examples:**

Test these alternating series for convergence.

a) it's easy to see that the denominator grows faster than the numerator so the terms are decreasing. Now we take the limit at infinity. , so series converges. |
b) again we can show the terms are decreasing, so we take the limit at infinity. , so series diverges. |

Once we've shown the series (without the alternating sign) is decreasing, we can use any of the limiting techniques to evaluate the limit. Sometimes we have to use l'Hopital's Rule.

For more on this topic, see the lesson on **Absolute and Conditional Convergence**.

**Practice:**

1) | 2) | 3) |

**Note:** in #2, the ( – 1)* ^{n}* is

**Solutions:**

In all 3 cases, it's easy to show that the terms are decreasing so we need only find limit at .

1) since the denominator approaches . Series converges by AST.

2) so we take limit at for which = 0. Series converges.

3) so we take limit at for which = 0. Series converges.

( calculus 2 index )

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