Alternating Series: Convergence, Remainders, and Classification

Learning Objectives

After reviewing this study guide, you should be able to:

• Use the Alternating Series Test to determine whether an alternating series converges.
• Estimate the sum of an alternating series and calculate the error bound using the remainder theorem.
• Explain and distinguish between absolute convergence and conditional convergence.

Key Terms & Glossary

• Alternating Series: A series whose terms alternate between positive and negative values.
• Alternating Series Test (AST): A convergence test specifically for alternating series based on decreasing term magnitude and a limit of zero.
• Partial Sum ($S_n): The sum of the first n terms of an infinite series.
• Absolute Convergence: A property of a series where the sum of the absolute values of its terms converges.
• Conditional Convergence: A condition where an alternating series converges, but the series of its absolute values diverges.
• Remainder (R_n): The error or difference between the true infinite sum S$and the$n$-th partial sum$S_n.

The "Big Idea"

So far, series analysis has primarily focused on positive terms. Alternating Series introduce terms that oscillate in sign (e.g., +, -, +, -, \dots). Because adding a negative term essentially subtracts from the accumulating total, alternating series have a "built-in" cancellation effect. This means an alternating series can converge much more easily than a series with strictly positive terms.

Understanding how these series converge—and whether their convergence relies solely on this cancellation effect (Conditional Convergence) or would happen regardless of signs (Absolute Convergence)—is foundational for mastering power series and Taylor series later in calculus.

Formula / Concept Box

[!NOTE] The Alternating Series Test can only prove convergence. If \lim_{n \to \infty} b_n \neq 0, the series diverges by the n-th Term Test for Divergence, NOT the Alternating Series Test.

Hierarchical Outline

• 1. Introduction to Alternating Series
  • Definition and standard forms.
  • The Alternating Harmonic Series vs. The Standard Harmonic Series.
• 2. The Alternating Series Test (AST)
  • Condition 1: Decreasing magnitudes (b_{n+1} \le b_n).
  • Condition 2: Limit approaches zero (\lim_{n \to \infty} b_n = 0).
• 3. Remainder and Error Estimation
  • Using partial sums (S_n) to approximate the true sum (S).
  • Bounding the error (|R_n| \le b_{n+1}$).
• 4. Absolute vs. Conditional Convergence
  • Absolute Convergence: Series converges even when all terms are positive.
  • Conditional Convergence: Series converges only because of alternating signs.

Visual Anchors

1. Classification of Convergence (Mermaid Flowchart)

2. The "Funnel" Effect of Partial Sums (TikZ Graph)

Caption: The partial sums of an alternating series oscillate above and below the true sum $S, squeezing closer with each step. This geometry is why the error |S - S_n| is always smaller than the next step b_{n+1}.

Definition-Example Pairs

Term: Alternating Harmonic Series Definition: The specific sequence \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots Real-World Example: Imagine tuning a guitar string where you overshoot the perfect pitch by 1 Hz, then undershoot by 0.5 Hz, then overshoot by 0.33 Hz, perpetually zeroing in on the correct note.

Term: Absolute Convergence Definition: A series \sum a_n is absolutely convergent if \sum |a_n| converges. Real-World Example: Tracking the total mileage on your car's odometer. Whether you drive forward or backward (positive or negative displacement), the total distance accumulated is a finite, absolute sum.

Term: Conditional Convergence Definition: A series converges, but the series of its absolute values diverges. Real-World Example: A tightrope walker taking steps left and right. If they alternate directions, they stay balanced near the center (converge). If they took all those steps in one direction (absolute value), they would fall off the rope (diverge).

Worked Examples

Example 1: Testing for Convergence

Problem: Determine if the series \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} converges or diverges. Step-by-Step Solution:

1. Identify b_n: The non-alternating part is b_n = \frac{1}{n}.
2. Check Condition 1 (Decreasing): Is b_{n+1} \le b_n$? Yes,$\frac{1}{n+1} < \frac{1}{n}$for all$n \ge 1.
3. Check Condition 2 (Limit): \lim_{n \to \infty} \frac{1}{n} = 0.
4. Conclusion: Since both conditions of the Alternating Series Test are met, the series converges.

Example 2: Estimating the Remainder

Problem: Approximate the sum of \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} using the 4th partial sum (S_4) and find the maximum error. Step-by-Step Solution:

1. Calculate S_4$: $S_4 = 1 - \frac{1}{4} + \frac{1}{9} - \frac{1}{16} = \frac{144 - 36 + 16 - 9}{144} = \frac{115}{144} \approx 0.7986$
2. Identify the Error Bound: The error $|R_4| \le b_5$.
3. Calculate $b_5$: $b_5 = \frac{1}{5^2} = \frac{1}{25} = 0.04$.
4. Conclusion: The approximation is $\approx 0.7986$, and we are guaranteed it is within $0.04 of the true infinite sum.

Example 3: Absolute vs. Conditional Convergence

Problem: Classify the convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}. Step-by-Step Solution:

1. Check Absolute Convergence: Take the absolute value to get \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$. This is a$p$-series with$p = 1/2$. Since$p \le 1, the absolute series diverges.
2. Check AST for Conditional Convergence:
   • Is b_n = \frac{1}{\sqrt{n}}$decreasing? Yes,$\sqrt{n+1} > \sqrt{n} \implies \frac{1}{\sqrt{n+1}} < \frac{1}{\sqrt{n}}$.
   • Is $\lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0$? Yes.
3. Conclusion: The series diverges absolutely but converges by the AST. Therefore, it is conditionally convergent.

Checkpoint Questions

1. Recall: What are the two specific conditions a series must meet to pass the Alternating Series Test?
2. Apply: If you want to estimate an alternating series to an error of less than $0.001, how do you determine which partial sum $S_n to stop at?
3. Distinguish: Can a series be absolutely convergent but fail to be conditionally convergent? Why or why not?
4. Analyze: If \lim_{n \to \infty} b_n = 5 in an alternating series, what test do you use to prove it diverges?

▶Answers to Checkpoint Questions (Click to expand)

1. The magnitudes of the terms must be decreasing (b_{n+1} \le b_n), and the limit of the terms must approach zero (\lim_{n \to \infty} b_n = 0).
2. Set the formula for the next term, b_{n+1}$, to be less than or equal to $0.001, and solve for$n.
3. Conditional convergence strictly means it converges but diverges when you take the absolute value. If it is absolutely convergent, it converges in both forms, so it by definition cannot be conditionally convergent.
4. You use the n-th Term Test for Divergence. Since the limit of the terms does not equal zero, the series must diverge.

Muddy Points & Cross-Refs

[!WARNING] Common Pitfall: A frequent mistake is assuming that if \lim_{n \to \infty} b_n \neq 0, the Alternating Series Test proves divergence. The AST can only prove convergence. If the limit is not zero, you must cite the n-th Term Test for Divergence to formally conclude the series diverges.

• Cross-Reference: The mechanics of absolute convergence will be critical when determining the Radius of Convergence for Power Series in future units. Keep these tests sharp!
• Cross-Reference: Remember the p$-series test (from the Integral Test module) when checking for Absolute Convergence. It's the fastest way to evaluate the absolute form of a fractional algebraic sequence.