What Is The Alternating Series Test

Alternating series, with their unique structure of alternating signs, require a specific test to determine their convergence. The alternating series test, also known as Leibniz's test, provides a straightforward method for determining whether an alternating series converges.

Understanding Alternating Series

An alternating series is an infinite series where the terms alternate in sign. It generally takes one of the following forms:

• ∑ (-1)^n * a_n = -a_1 + a_2 - a_3 + a_4 - ...
• ∑ (-1)^(n+1) * a_n = a_1 - a_2 + a_3 - a_4 + ...

Where a_n is a positive real number for all n. The key characteristic is the (-1)^n or (-1)^(n+1) term, which ensures the alternating signs.

Key Components

Before diving into the test itself, let's clarify the crucial components:

• a_n: This is the magnitude of the nth term. It is essential that a_n is positive.
• Alternating Sign: The (-1)^n or (-1)^(n+1) dictates the sign of each term, ensuring the alternating behavior.

Examples of Alternating Series

To better understand alternating series, consider these examples:

1. 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... (Alternating Harmonic Series)
2. -1 + 1/4 - 1/9 + 1/16 - 1/25 + ...
3. 2 - 2/3 + 2/9 - 2/27 + 2/81 - ...

The Alternating Series Test (Leibniz's Test)

The alternating series test, credited to Gottfried Wilhelm Leibniz, provides conditions under which an alternating series converges. The test states:

Given an alternating series ∑ (-1)^n * a_n or ∑ (-1)^(n+1) * a_n, where a_n > 0 for all n, the series converges if the following two conditions are met:

1. a_n is a decreasing sequence: That is, a_(n+1) ≤ a_n for all n greater than some integer N. In simpler terms, the magnitude of the terms must decrease (or at least not increase) as n increases.
2. The limit of a_n as n approaches infinity is zero: lim (n→∞) a_n = 0. The terms must get infinitesimally small as the series progresses.

If both conditions are satisfied, the alternating series is convergent. If either condition fails, the test is inconclusive, and other convergence tests must be considered.

Steps to Apply the Alternating Series Test

Here's a step-by-step guide on how to apply the alternating series test:

1. Verify the Alternating Series Form: Confirm that the series is indeed alternating, meaning the signs alternate due to a (-1)^n or (-1)^(n+1) term.

2. Identify a_n: Isolate the positive term a_n from the series.

3. Check if a_n is Decreasing: Determine whether a_n is a decreasing sequence. This can be done by:

   • Direct Comparison: Compare consecutive terms, i.e., show that a_(n+1) ≤ a_n for all n (or at least for n greater than some N).
   • Using Derivatives: If a_n can be expressed as a function f(x), show that the derivative f'(x) is negative for x greater than some N. This indicates that the function (and thus the sequence) is decreasing.

4. Evaluate the Limit: Calculate the limit of a_n as n approaches infinity: lim (n→∞) a_n.

5. Apply the Test:

   • If both conditions are met (a_n is decreasing and the limit is zero), the alternating series converges.
   • If either condition fails, the test is inconclusive.

Examples of Applying the Alternating Series Test

Let's illustrate the use of the alternating series test with several examples:

Example 1: The Alternating Harmonic Series

Consider the alternating harmonic series:

∑ (-1)^(n+1) / n = 1 - 1/2 + 1/3 - 1/4 + 1/5 - ...

1. Alternating Series Form: The series is alternating due to the (-1)^(n+1) term.

2. a_n: a_n = 1/n

3. a_n is Decreasing: To show that a_n is decreasing, we can compare consecutive terms:

   1/(n+1) ≤ 1/n for all n ≥ 1. This is true since the denominator increases, making the fraction smaller.

4. Evaluate the Limit:

   lim (n→∞) 1/n = 0

5. Apply the Test: Both conditions are met, so the alternating harmonic series converges.

Example 2: A Slightly Modified Series

Consider the series:

∑ (-1)^n * (n / (n^2 + 1)) = -1/2 + 2/5 - 3/10 + 4/17 - ...

1. Alternating Series Form: The series is alternating due to the (-1)^n term.

2. a_n: a_n = n / (n^2 + 1)

3. a_n is Decreasing: To show that a_n is decreasing, we can use derivatives. Let f(x) = x / (x^2 + 1). Then:

   f'(x) = ((x^2 + 1) * 1 - x * 2x) / (x^2 + 1)^2 = (1 - x^2) / (x^2 + 1)^2

   For x > 1, f'(x) is negative, indicating that f(x) is decreasing. Therefore, a_n is decreasing for n > 1.

4. Evaluate the Limit:

   lim (n→∞) n / (n^2 + 1) = 0 (by dividing numerator and denominator by n^2)

5. Apply the Test: Both conditions are met, so the series converges.

Example 3: A Divergent Alternating Series

Consider the series:

∑ (-1)^(n+1) * (n / (2n + 1)) = 1/3 - 2/5 + 3/7 - 4/9 + ...

1. Alternating Series Form: The series is alternating due to the (-1)^(n+1) term.

2. a_n: a_n = n / (2n + 1)

3. a_n is Decreasing: Let's analyze if a_(n+1) ≤ a_n

   (n+1) / (2(n+1) + 1) ≤ n / (2n + 1) (n+1) / (2n + 3) ≤ n / (2n + 1) (n+1)(2n+1) ≤ n(2n+3) 2n^2 + 3n + 1 ≤ 2n^2 + 3n 1 ≤ 0 (False)

   The sequence is not decreasing.

4. Evaluate the Limit:

   lim (n→∞) n / (2n + 1) = 1/2 ≠ 0

5. Apply the Test: Neither condition is met. The limit is not zero, and the sequence is not decreasing. Therefore, the alternating series test fails, and we can conclude that the series diverges (since the limit of the terms is not zero).

Absolute vs. Conditional Convergence

When dealing with alternating series, it's important to understand the concepts of absolute and conditional convergence:

• Absolute Convergence: A series ∑ (-1)^n * a_n converges absolutely if the series ∑ |a_n| converges. In other words, if you take the absolute value of each term and the resulting series converges, the original alternating series converges absolutely.
• Conditional Convergence: A series ∑ (-1)^n * a_n converges conditionally if it converges (according to the alternating series test), but the series ∑ |a_n| diverges.

Example: Absolute vs. Conditional Convergence

• Alternating Harmonic Series: ∑ (-1)^(n+1) / n = 1 - 1/2 + 1/3 - 1/4 + ...

  • We know this series converges by the alternating series test.
  • However, the series ∑ |1/n| = 1 + 1/2 + 1/3 + 1/4 + ... is the harmonic series, which is known to diverge.
  • Therefore, the alternating harmonic series converges conditionally.

• Series: ∑ (-1)^n / n^2 = -1 + 1/4 - 1/9 + 1/16 - ...

  • This series converges by the alternating series test.
  • The series ∑ |1/n^2| = 1 + 1/4 + 1/9 + 1/16 + ... converges (it's a p-series with p = 2 > 1).
  • Therefore, the series ∑ (-1)^n / n^2 converges absolutely.

Why Does Absolute/Conditional Convergence Matter?

• Rearrangements: Absolutely convergent series can be rearranged without changing their sum. Conditionally convergent series, however, can be rearranged to converge to any value, or even diverge! This makes absolute convergence a stronger and more desirable property.
• Operations: Certain operations, like multiplying two infinite series together, are only guaranteed to work correctly if at least one of the series converges absolutely.

Common Mistakes to Avoid

When using the alternating series test, be aware of these common pitfalls:

1. Forgetting to Check Both Conditions: It's crucial to verify both that a_n is decreasing and that lim (n→∞) a_n = 0. Failing to check one of these conditions can lead to incorrect conclusions.
2. Assuming Decreasing Behavior Too Quickly: Make sure to rigorously prove that a_n is decreasing, especially for more complex series. Simply observing the first few terms might be misleading. Use derivatives or direct comparison to provide a solid argument.
3. Misinterpreting the Inconclusive Result: If either condition fails, the alternating series test is inconclusive. This does not mean the series diverges; it simply means you need to use a different convergence test.
4. Ignoring the Alternating Requirement: The series must be strictly alternating in sign to apply this test. If the signs don't alternate, the test is not applicable.
5. Confusing a_n with the Entire Term: Remember that a_n represents the magnitude of the terms (i.e., the positive part). Don't include the (-1)^n or (-1)^(n+1) part when determining a_n.
6. Incorrectly Calculating the Limit: Ensure that you correctly evaluate lim (n→∞) a_n. Use appropriate techniques like L'Hôpital's Rule or algebraic manipulation when necessary.

Beyond the Alternating Series Test

While the alternating series test is powerful for alternating series, it's just one tool in the arsenal of convergence tests. Here are some other tests that you might need to use:

• Divergence Test (nth Term Test): If lim (n→∞) a_n ≠ 0, then the series ∑ a_n diverges. This is usually the first test to apply because it's simple and can quickly identify divergent series.
• Integral Test: If f(x) is a continuous, positive, and decreasing function on [1, ∞), and a_n = f(n), then the series ∑ a_n and the integral ∫[1, ∞) *f(x) dx either both converge or both diverge.
• Comparison Test: If 0 ≤ a_n ≤ b_n for all n, then:
  • If ∑ b_n converges, then ∑ a_n converges.
  • If ∑ a_n diverges, then ∑ b_n diverges.
• Limit Comparison Test: If lim (n→∞) (a_n / b_n) = c, where 0 < c < ∞, then ∑ a_n and ∑ b_n either both converge or both diverge.
• Ratio Test: Let L = lim (n→∞) |a_(n+1) / a_n|. Then:
  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.
• Root Test: Let L = lim (n→∞) |a_n|^(1/n). Then:
  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.

The choice of which test to use depends on the specific series you're analyzing. Experience and pattern recognition are key to selecting the most appropriate test.

Real-World Applications

While the alternating series test might seem purely theoretical, it has applications in various fields:

• Physics: In quantum mechanics, alternating series can arise in perturbation theory when calculating energy levels of systems.
• Engineering: Signal processing and control systems often involve infinite series, and determining their convergence is crucial for stability analysis.
• Computer Science: Approximating functions using Taylor series (which can be alternating) is common in numerical analysis and scientific computing.
• Economics: Certain economic models involve infinite sums, and understanding their convergence is important for making predictions and analyzing stability.

Conclusion

The alternating series test is a valuable tool for determining the convergence of alternating series. By verifying that the magnitude of the terms is decreasing and that the limit of the terms approaches zero, you can confidently conclude that the series converges. Understanding the difference between absolute and conditional convergence adds further nuance to your analysis. Remember to avoid common mistakes and to consider other convergence tests when the alternating series test is inconclusive. With practice and careful application, you can master the alternating series test and expand your understanding of infinite series.