Does The Alternating Series Test Prove Absolute Convergence

Does the Alternating Series Test Prove Absolute Convergence?

The short answer is: no, the alternating series test does not prove absolute convergence. It only proves conditional convergence. This subtle but crucial difference is often a source of confusion for students of calculus. This article will clarify the distinction and explain why.

The alternating series test is a valuable tool for determining the convergence of series that alternate in sign. It states that if a series has the form ∑ (-1)^n * b_n, where b_n ≥ 0 for all n, and b_n is a monotonically decreasing sequence approaching zero as n approaches infinity (lim (n→∞) b_n = 0), then the series converges.

However, this test only guarantees conditional convergence. This means the series converges, but if you take the absolute value of each term (making all terms positive), the resulting series ∑ |(-1)^n * b_n| = ∑ b_n might diverge.

Understanding the Difference: Conditional vs. Absolute Convergence

• Absolute Convergence: A series ∑ a_n is absolutely convergent if the series of absolute values, ∑ |a_n|, converges. Absolutely convergent series are always convergent.

• Conditional Convergence: A series ∑ a_n is conditionally convergent if it converges, but the series of absolute values, ∑ |a_n|, diverges. This means the series converges only because of the alternating signs; removing the alternating signs would cause the series to diverge.

Why the Alternating Series Test Doesn't Guarantee Absolute Convergence

The alternating series test relies on the cancellation of positive and negative terms to achieve convergence. It exploits the alternating nature to create a telescoping effect, where terms partially cancel each other out. If you remove this alternating pattern by taking the absolute value of each term, this cancellation effect disappears, and the series may diverge. The series might not converge if all terms are positive.

Example Illustrating the Difference

Consider the alternating harmonic series: ∑ (-1)^(n+1) / n = 1 - 1/2 + 1/3 - 1/4 + ...

The alternating series test shows this series converges (it's a classic example). However, if we take the absolute value of each term, we get the harmonic series: ∑ 1/n = 1 + 1/2 + 1/3 + 1/4 + ... which is known to diverge. Therefore, the alternating harmonic series is conditionally convergent.

In Conclusion

The alternating series test is a powerful tool, but it's essential to remember its limitations. It proves convergence, but only conditional convergence. To determine absolute convergence, you need to apply different tests, such as the comparison test, limit comparison test, integral test, or ratio test, to the series of absolute values. Understanding this distinction is key to accurately analyzing the convergence of infinite series. Always be mindful of the specific conditions and implications of each convergence test you apply.