Question

77–87. Absolute or conditional convergenceDetermine whether the following series converge absolutely, converge conditionally, or diverge.∑ (from k = 1 to ∞) (−2)ᵏ⁺¹ / k²

Accepted Answer

Identify the given series: $$ \sum_{k=1}^{\infty} \frac{(-2)^{k+1}}{k^2} . $$Notice that the terms involve $$(-2)^{k+1}$$, which grows exponentially in magnitude, and the denominator $$k^2$$, which grows polynomially. To analyze convergence, first consider the absolute value of the terms: $$ \left| \frac{(-2)^{k+1}}{k^2} \right| = \frac{2^{k+1}}{k^2} . $$This transforms the series into $$ \sum_{k=1}^{\infty} \frac{2^{k+1}}{k^2} . $$Check for absolute convergence by testing the series $$ \sum_{k=1}^{\infty} \frac{2^{k+1}}{k^2} . $$Since $$2^{k+1}$$ grows exponentially and $$k^2$$ grows polynomially, the terms do not approach zero fast enough for convergence. Use the root test or ratio test to confirm this behavior. Next, consider conditional convergence by analyzing the original series with alternating signs. However, the factor $$(-2)^{k+1}$$ does not produce a simple alternating sign pattern because the base is $$-2$$, not $$-1. $$The terms actually grow in magnitude exponentially, so the series does not satisfy the conditions for the Alternating Series Test. Conclude that since the series does not converge absolutely (step 3) and does not converge conditionally (step 4), the series diverges.

77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²

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Key Concepts

Absolute Convergence

Conditional Convergence

Alternating Series Test

77–87. Absolute or conditional convergenceDetermine whether the following series converge absolutely, converge conditionally, or diverge.∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²