Alternating Series Test- Slader Solutions and Examples

What the Alternating Series Test Actually Is

The Alternating Series Test (also called the Leibniz Test) tells you when an alternating series converges. That's it. No extra magic.

An alternating series looks like this:

∑ (-1)ⁿ aₙ = a₁ - a₂ + a₃ - a₄ + ...

or

∑ (-1)ⁿ⁺¹ aₙ = -a₁ + a₂ - a₃ + a₄ - ...

The terms just switch signs. One positive, one negative, one positive, and so on.

The Two Conditions You Must Check

For an alternating series ∑ (-1)ⁿ aₙ to converge, two things need to be true:

• Monotonicity: The sequence {aₙ} must eventually decrease. That means a₁ ≥ a₂ ≥ a₃ ≥ ... ≥ aₙ ≥ ...
• Limit to zero: The limit of aₙ as n approaches infinity must equal zero. In math: lim(n→∞) aₙ = 0

Both conditions are required. Not one or the other. Both.

The Test in Plain English

Think of it this way: you're adding positive and negative terms that get smaller and smaller, eventually approaching zero. The positive terms push the sum up, the negative terms push it down, and the pushes get smaller each time. The sum gets trapped in a narrower and narrower range until it settles on a single number.

That's convergence.

Step-by-Step: How to Apply the Test

Step 1: Identify the positive part

Extract aₙ from your alternating series. Remove the (-1)ⁿ part. You're left with just the positive sequence.

Step 2: Check the limit

Calculate lim(n→∞) aₙ. If this limit is not zero, the series diverges. Done. No need to check anything else.

Step 3: Check monotonic decrease

Show that aₙ₊₁ ≤ aₙ for all n (or for all n beyond some point). You can do this by:

• Computing a few terms and seeing the pattern
• Using derivatives if aₙ is a function
• Solving the inequality aₙ₊₁ ≤ aₙ directly

Step 4: State your conclusion

If both conditions hold, the alternating series converges. If either fails, it diverges.

Examples That Actually Help

Example 1: The Alternating Harmonic Series

Test: ∑ from n=1 to ∞ of (-1)ⁿ⁺¹ (1/n)

This is: 1 - 1/2 + 1/3 - 1/4 + 1/5 - ...

Step 1: aₙ = 1/n

Step 2: lim(n→∞) 1/n = 0 ✓

Step 3: Is 1/(n+1) ≤ 1/n? Yes. The sequence decreases.

Step 4: Converges. This series converges to ln(2).

Example 2: A Diverging Alternating Series

Test: ∑ from n=1 to ∞ of (-1)ⁿ (1 + 1/n)

Step 1: aₙ = 1 + 1/n

Step 2: lim(n→∞) (1 + 1/n) = 1. Not zero.

Step 3: Doesn't matter. Already diverges.

The terms don't approach zero, so the series cannot converge.

Example 3: Alternating with a Factorial

Test: ∑ from n=0 to ∞ of (-1)ⁿ (3ⁿ / n!)

Step 1: aₙ = 3ⁿ / n!

Step 2: lim(n→∞) 3ⁿ / n! = 0. Factorials grow faster than exponentials. ✓

Step 3: The ratio aₙ₊₁ / aₙ = 3/(n+1). For n ≥ 2, this ratio is less than 1, so the sequence decreases.

Step 4: Converges. This series converges to e⁻³.

Where Students Actually Go Wrong

• Forgetting to check the limit: This is the most common mistake. If lim aₙ ≠ 0, the series diverges immediately. No need to check monotonicity.
• Assuming monotonicity from the start: The sequence must decrease eventually, not necessarily from n=1. But if it starts increasing after n=1000, that's a problem.
• Confusing with other tests: The Alternating Series Test only works for alternating series. Don't try to force it on non-alternating series.
• Wrong sign pattern: Make sure you correctly identify (-1)ⁿ vs (-1)ⁿ⁺¹. One starts positive, one starts negative.

The Remainder Estimate: How Close Are You?

When an alternating series converges, the error when approximating by the first N terms is at most the magnitude of the next term.

|Rₙ| ≤ aₙ₊₁

This is called the Alternating Series Estimation Theorem.

Example: The alternating harmonic series converges to ln(2) ≈ 0.6931. If you sum the first 4 terms: 1 - 1/2 + 1/3 - 1/4 = 0.5833. The error is at most |1/5| = 0.2. The actual error is about 0.1098, which is less than 0.2.

Using Slader for Solutions

Slader provides step-by-step solutions to textbook problems. For alternating series test problems, here's what to look for:

• Search your textbook problem number
• Check if the solution breaks down each condition separately
• Look for where they calculate the limit
• See how they prove monotonicity

Slader works for verification. It does not replace understanding the method. If you copy answers without knowing why the test works, you'll fail the exam.

Alternating Series Test vs Other Convergence Tests

The Alternating Series Test is narrow in scope. It only applies when you have alternating signs. For everything else, you need other tools.

Getting Started: Your Checklist

When you encounter a series problem and need to test for convergence:

1. Look at the series. Does it have (-1)ⁿ or (-1)ⁿ⁺¹? If yes, try the Alternating Series Test first.
2. Extract aₙ. Calculate lim(n→∞) aₙ. If ≠ 0, the series diverges.
3. If the limit equals zero, prove the sequence decreases.
4. State your conclusion.
5. Use the remainder estimate if the problem asks for approximation accuracy.

That's the entire process. Practice it until it's automatic.