Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]

검증된 단계별 안내

Identify the given series: \( \sum_{n=1}^{\infty} (-1)^{n+1} \sqrt[n]{10} \). This is an alternating series because of the factor \( (-1)^{n+1} \).

Check the absolute convergence by considering the series of absolute values: \( \sum_{n=1}^{\infty} \left| (-1)^{n+1} \sqrt[n]{10} \right| = \sum_{n=1}^{\infty} \sqrt[n]{10} \).

Analyze the behavior of the terms \( \sqrt[n]{10} = 10^{1/n} \). As \( n \to \infty \), \( 10^{1/n} \to 1 \), so the terms do not approach zero.

Since the terms of the absolute value series do not approach zero, the series \( \sum_{n=1}^{\infty} \sqrt[n]{10} \) diverges, so the original series does not converge absolutely.

Next, apply the Alternating Series Test to the original series: check if the terms \( \sqrt[n]{10} \) decrease monotonically to zero. Since they approach 1, not zero, the Alternating Series Test fails, so the series does not converge conditionally either.

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주요 개념

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Absolute Convergence

A series ∑a_n converges absolutely if the series of absolute values ∑|a_n| converges. Absolute convergence guarantees convergence regardless of the order of terms and is a stronger form of convergence than conditional convergence.

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_n converges, but ∑|a_n| diverges. Alternating series often exhibit conditional convergence, relying on the alternating signs and decreasing terms.

Root Test for Convergence

The root test uses the limit L = lim (n→∞) ⁿ√|a_n| to determine convergence: if L < 1, the series converges absolutely; if L > 1, it diverges; if L = 1, the test is inconclusive. This test is especially useful for series involving nth roots.

Absolute and Conditional ConvergenceWhich of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]