Question

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 / (n² + 2n + 1)]

Accepted Answer

Identify the given series: $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2 + 2n + 1} . $$Notice that the denominator can be factored as $$ (n+1)^2 . $$Check for absolute convergence by considering the series of absolute values: $$ \sum_{n=1}^{\infty} \left| \frac{(-1)^{n-1}}{(n+1)^2} ight| = \sum_{n=1}^{\infty} \frac{1}{(n+1)^2} . $$This is a p-series with $$ p = 2 $$, which is known to converge. Since the series of absolute values converges, the original series converges absolutely. Absolute convergence implies convergence. 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 $$ n 	o \infty $$, satisfying the conditions of the Alternating Series Test. Summarize: The series converges absolutely (and hence converges) because the absolute value series is a convergent p-series with $$ p \u003e 1 .$$

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)]

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Key Concepts

Absolute Convergence

Conditional Convergence

Alternating Series Test