Alternating Series and the Alternating Series Test

What an Alternating Series Is

An alternating series is an infinite series whose terms alternate between positive and negative values. The signs typically follow a pattern such as positive, negative, positive, negative, and so on.

A common way to write an alternating series is:

The factor creates the alternating signs. When is even, the term is positive; when is odd, the term is negative.

For example:

The signs alternate, but that alone does not tell us whether the series converges. We need a test.

The Alternating Series Test

The Alternating Series Test gives us a direct way to determine whether an alternating series converges.

Consider a series of the form:

The series converges if both of the following conditions are satisfied:

The terms eventually decrease as increases.
.

If both conditions hold, the series converges.

Notice something important: the terms do not have to become zero immediately. They only need to approach zero in the limit, and they must eventually decrease.

A Step-by-Step Method

Check that the signs alternate. Look for a factor such as .
Define the positive term. Identify , the magnitude of each term.
Check that the terms decrease. Determine whether eventually decreases.
Check the limit. Evaluate .
State the conclusion clearly. If both conditions are satisfied, the series converges by the Alternating Series Test.

Worked Examples

Example 1: The Alternating Harmonic Series

Determine whether the following series converges:

Step 1: Identify the positive term:

Step 2: Check whether the terms decrease. Since decreases as increases, the first condition is satisfied.

Step 3: Evaluate the limit:

Step 4: Both conditions are satisfied.

Result: The series converges by the Alternating Series Test.

Example 2: Alternating Powers

Determine whether the following series converges:

Step 1: Identify .

Step 2: The sequence decreases because the denominator increases with .

Step 3: Evaluate the limit:

Result: The series converges by the Alternating Series Test.

Example 3: An Alternating Series with a Logarithm

Determine whether the following series converges:

Step 1: Let .

Step 2: The sequence decreases for sufficiently large because the denominator grows faster than the logarithm.

Step 3: Evaluate the limit:

Result: The series converges by the Alternating Series Test.

Example 4: When the Test Fails

Consider the series:

Step 1: The signs alternate, so it looks like an alternating series.

Step 2: Let . However, the terms do not approach zero.

Step 3: .

Result: The series diverges because its terms do not approach zero.

Decreasing Terms: What Does That Mean?

One of the most important parts of the Alternating Series Test is showing that the positive terms are decreasing.

In many AP Calculus problems, this is obvious from the expression. For example:

As increases, the denominator increases, so the fraction gets smaller. Therefore the terms decrease.

Sometimes, however, the expression is more complicated. In that situation, you can use derivatives to show that the corresponding function is decreasing.

If for a continuous function , then is decreasing on that interval.

Why the Limit Must Be Zero

Every convergent infinite series must have terms that approach zero. This is not unique to alternating series.

Therefore, if you calculate and get a nonzero number, or if the limit does not exist, you can immediately conclude that the series diverges.

This is often the fastest way to rule out convergence.

Common Mistakes

Assuming every alternating series converges

Alternating signs are not enough. The terms must also decrease toward zero. A series can alternate forever and still diverge.

Forgetting the limit condition

Before doing anything complicated, check . If that limit is not zero, the series diverges immediately.

Showing only that the terms get small

Getting smaller is not enough by itself. The Alternating Series Test requires the positive terms to eventually decrease and to approach zero.

Saying "the series converges" without naming the test

On an AP free-response question, write the reasoning explicitly: the terms decrease and approach zero, so the series converges by the Alternating Series Test.

Alternating Series Test vs. Other Tests

The Alternating Series Test is especially useful when the signs alternate. But it is not always the strongest test available.

Alternating Series Test: use it when the signs alternate and the positive terms decrease toward zero.
Geometric Series Test: use it when the series has a constant ratio between consecutive terms.
p-Series Test: use it for series of the form .
Ratio Test: useful when factorials, powers, or complicated products appear.
Comparison Tests: useful when a positive series resembles a familiar convergent or divergent series.

On the AP exam, the goal is not to use the fanciest test. The goal is to recognize the structure of the series and use an appropriate test with a complete justification.

Practice Problems

Question 1

Determine whether the series converges or diverges:

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Question 2

Determine whether the series converges or diverges:

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Question 3

Determine whether the series converges or diverges:

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Question 4 (Justification)

Explain why the following alternating series converges:

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Question 5 (Test Selection)

Explain why the Alternating Series Test can be applied to:

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