Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / (2√k − 1)

Accepted Answer

Identify the given series: $$ \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2\sqrt{k} - 1} . $$This is an alternating series because of the factor $$ (-1)^{k+1} $$, which alternates the sign of each term. Check for absolute convergence by considering the series of absolute values: $$ \sum_{k=1}^{\infty} \left| \frac{(-1)^{k+1}}{2\sqrt{k} - 1} ight| = \sum_{k=1}^{\infty} \frac{1}{2\sqrt{k} - 1} . $$Determine if this series converges. To analyze the absolute value series, compare $$ \frac{1}{2\sqrt{k} - 1}  to a $$simpler series. Since for large $$ k $$, $$ 2\sqrt{k} - 1 \approx 2\sqrt{k} $$, compare it to $$ \frac{1}{\sqrt{k}} $$, which is a p-series with $$ p = \frac{1}{2} \u003c 1 $$ and diverges. Since the absolute value series diverges, the original series does not converge absolutely. Next, apply the Alternating Series Test to the original series by checking if the terms $$ b_k = \frac{1}{2\sqrt{k} - 1} $$ decrease monotonically to zero. Verify that $$ b_k  is $$positive, decreasing, and $$ \lim_{k 	o \infty} b_k = 0 . If $$these conditions hold, the series converges conditionally by the Alternating Series Test.

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / (2√k − 1)

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Key Concepts

Absolute Convergence

Conditional Convergence

Alternating Series Test