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 / (n² + 2n + 1)]

Verified step by step guidance

1. Identify the given series: \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2 + 2n + 1} \). Notice that the denominator can be factored as \( (n+1)^2 \).
2. Check for absolute convergence by considering the series of absolute values: \( \sum_{n=1}^{\infty} \left| \frac{(-1)^{n-1}}{(n+1)^2} \right| = \sum_{n=1}^{\infty} \frac{1}{(n+1)^2} \). This is a p-series with \( p = 2 \), which is known to converge.
3. Since the series of absolute values converges, the original series converges absolutely. Absolute convergence implies convergence.
4. For completeness, recognize that the original series is an alternating series because of the factor \( (-1)^{n-1} \). The terms \( \frac{1}{(n+1)^2} \) decrease to zero as \( n \to \infty \), satisfying the conditions of the Alternating Series Test.
5. Summarize: The series converges absolutely (and hence converges) because the absolute value series is a convergent p-series with \( p > 1 \).

Key Concepts

Absolute Convergence

A series ∑aₙ converges absolutely if the series of absolute values ∑|aₙ| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it often simplifies analysis by allowing the use of comparison tests on positive terms.

Conditional Convergence

A series ∑aₙ converges conditionally if it converges, but does not converge absolutely. This means the series converges due to the alternating signs or specific term behavior, but the series of absolute values diverges, highlighting the importance of sign alternation.

Alternating Series Test

The Alternating Series Test states that an alternating series ∑(-1)ⁿbₙ converges if the sequence bₙ is positive, decreasing, and approaches zero. This test helps determine conditional convergence when absolute convergence fails.