Textbook Question
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k³⁄⁷
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14. Sequences & Series
Convergence Tests
2:18 minutesProblem 10.4.53
Textbook Question
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) k / √(k² + 1)
Verified step by step guidance1
First, write down the general term of the series: \(a_k = \frac{k}{\sqrt{k^2 + 1}}\).
To analyze convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Simplify the expression inside the square root to understand the dominant terms.
Divide numerator and denominator by \(k\) to rewrite \(a_k\) as \(\frac{k}{\sqrt{k^2 + 1}} = \frac{k}{k \sqrt{1 + \frac{1}{k^2}}} = \frac{1}{\sqrt{1 + \frac{1}{k^2}}}\).
Evaluate the limit \(\lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{1}{\sqrt{1 + \frac{1}{k^2}}}\) to determine if the terms approach zero, which is necessary for convergence.
Since the terms do not approach zero, conclude that the series diverges by the Test for Divergence (also known as the nth-term test).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence and Divergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.
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Limit Comparison Test
The Limit Comparison Test compares a given series with a known benchmark series by examining the limit of their term ratios. If the limit is a positive finite number, both series either converge or diverge together. This test is useful when terms resemble a simpler series.
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Behavior of Terms and nth-Term Test for Divergence
The nth-term test states that if the limit of the terms of a series does not approach zero, the series must diverge. Analyzing the behavior of the general term, especially for large n, helps quickly identify divergence without further tests.
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