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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 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

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1
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.
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Divergence Test (nth Term Test)

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.
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Integral Test

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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P-Series and Harmonic Series
Related Practice
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