Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)k! / (kᵏ + 3)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.7.49
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ / √(k³ᐟ² + k)
Verified step by step guidance1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \). Notice it is an alternating series because of the factor \( (-1)^k \).
Check for absolute convergence by considering the absolute value of the terms: \( \left| \frac{(-1)^k}{\sqrt{k^{3/2} + k}} \right| = \frac{1}{\sqrt{k^{3/2} + k}} \).
Simplify the denominator inside the square root to understand the behavior for large \( k \): \( \sqrt{k^{3/2} + k} = \sqrt{k^{3/2}(1 + k^{1/2 - 1})} = \sqrt{k^{3/2}} \sqrt{1 + k^{-1/2}} \). Since \( \sqrt{k^{3/2}} = k^{3/4} \), the term behaves like \( \frac{1}{k^{3/4}} \) for large \( k \).
Determine if the series of absolute values \( \sum \frac{1}{k^{3/4}} \) converges. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Here, \( p = \frac{3}{4} < 1 \), so the series does not converge absolutely.
Since the series is alternating, apply the Alternating Series Test: check if the terms \( b_k = \frac{1}{\sqrt{k^{3/2} + k}} \) decrease monotonically to zero. If they do, the series converges conditionally.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute and Conditional Convergence
Absolute convergence occurs when the series of absolute values converges, ensuring the original series converges regardless of term signs. Conditional convergence happens when the series converges, but the series of absolute values diverges. Distinguishing between these helps classify the behavior of alternating series.
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07:51
Choosing a Convergence Test
Alternating Series Test
The Alternating Series Test determines convergence of series whose terms alternate in sign. If the absolute value of terms decreases monotonically to zero, the series converges. This test is useful for series with factors like (−1)^k, as in the given problem.
Recommended video:
10:54Alternating Series Test
Comparison and p-Series Tests
Comparison tests involve comparing a given series to a known benchmark series to determine convergence. The p-series test states that ∑ 1/k^p converges if p > 1 and diverges otherwise. Recognizing the dominant term in the denominator helps apply these tests effectively.
Recommended video:
04:30P-Series and Harmonic Series
Related Practice
Textbook Question
21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 2aₙ; a₁ = 2
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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 ∞) 1 / ( (3k + 1)(3k + 4) )
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Textbook Question
8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.
∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4
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Textbook Question
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 3 to ∞) 5 / (2 + lnk)
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