Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)
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 10.8.29
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (10ᵏ + 1) / k¹⁰
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Identify the general term of the series: \(a_k = \frac{10^k + 1}{k^{10}}\).
Observe the behavior of the numerator and denominator separately: the numerator grows exponentially as \$10^k\(, while the denominator grows polynomially as \)k^{10}$.
Recall that exponential growth dominates polynomial growth, so \$10^k\( grows much faster than \)k^{10}$ as \(k \to \infty\).
Apply the Divergence Test (also known as the Test for Divergence) by checking the limit of \(a_k\) as \(k\) approaches infinity: calculate \(\lim_{k \to \infty} \frac{10^k + 1}{k^{10}}\).
Since the numerator grows exponentially and the denominator polynomially, this limit does not approach zero; therefore, by the Divergence Test, the series diverges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity.
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06:52
Convergence of an Infinite Series
Comparison Test
The Comparison Test involves comparing a given series to a second series with known convergence behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.
Recommended video:
09:25Direct Comparison Test
Behavior of Exponential vs. Polynomial Terms
Exponential terms like 10^k grow much faster than polynomial terms like k^10. When analyzing series terms involving both, the exponential growth dominates, often causing divergence despite the polynomial denominator.
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5:46Graphs of Exponential Functions
Related Practice
Textbook Question
Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.
Textbook Question
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 k (2ᵏ⁺¹ / (9ᵏ − 1))
Textbook Question
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)
Textbook Question
9–15. Geometric sums Evaluate each geometric sum.
∑ k = 0 to 83ᵏ
1
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)