Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.4.35

Chapter 10, Problem 10.4.35

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) (k / (k + 10))ᵏ

Verified step by step guidance

First, identify the general term of the series: \(a_k = \left( \frac{k}{k + 10} \right)^k\).

Apply the Divergence Test by finding the limit of \(a_k\) as \(k\) approaches infinity: compute \(\lim_{k \to \infty} \left( \frac{k}{k + 10} \right)^k\).

Rewrite the term inside the limit to a form that is easier to analyze: \(\left( \frac{k}{k + 10} \right)^k = \left( 1 - \frac{10}{k + 10} \right)^k\).

Recognize that this limit resembles the form \(\lim_{n \to \infty} \left( 1 - \frac{c}{n} \right)^n = e^{-c}\) for some constant \(c\), and use this to evaluate the limit.

If the limit of \(a_k\) is not zero, conclude by the Divergence Test that the series diverges; if the limit is zero, consider applying other tests such as the Root Test or Ratio Test for further analysis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. It is a quick initial check to determine if a series cannot converge, but if the limit is zero, the test is inconclusive.

Integral Test

The Integral Test relates the convergence of a series to the convergence of an improper integral. If a function f(k) is positive, continuous, and decreasing for k ≥ 1, then the series ∑f(k) and the integral ∫f(x)dx from 1 to infinity either both converge or both diverge.

p-Series Test

A p-series is a series of the form ∑ 1/k^p. It converges if and only if p > 1 and diverges otherwise. This test helps quickly determine convergence for series resembling p-series or can be used as a comparison benchmark.

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Related Practice

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{tan⁻¹(n)}

Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ · (k!) / (kᵏ)(Hint: Show that k! / kᵏ ≤ 2 / k², for k ≥ 3.)

Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{ⁿ√(e³ⁿ⁺⁴)}

views

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 ∞) 1 / (k³ᐟ² + 1)

Textbook Question

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.∑ (k = 1 to ∞) (k / (k + 10))ᵏ

Verified video answer for a similar problem:

Key Concepts

Divergence Test

Integral Test

p-Series Test

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) (k / (k + 10))ᵏ