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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 10.8.3
Chapter 10, Problem 10.8.3

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 = 3 to ∞) (2k²) / (k² − k − 2)

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First, examine the general term of the series: \(\frac{2k^{2}}{k^{2} - k - 2}\). To understand its behavior, try to simplify the denominator by factoring it.
Factor the quadratic expression in the denominator: \(k^{2} - k - 2 = (k - 2)(k + 1)\). So the term becomes \(\frac{2k^{2}}{(k - 2)(k + 1)}\).
Next, analyze the behavior of the term for large \(k\). Since the numerator and denominator are both degree 2 polynomials, consider the limit of the term as \(k \to \infty\) to compare it with a simpler series.
Because the degrees of numerator and denominator are the same, the term behaves like a constant times \(\frac{k^{2}}{k^{2}}\), which simplifies to a constant. This suggests the terms do not approach zero, which is a key condition for convergence.
Based on this, the Divergence Test (also called the Test for Divergence) is the appropriate convergence test to apply first, since if the terms do not approach zero, the series cannot converge.

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Key Concepts

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

Series and Convergence

A series is the sum of the terms of a sequence. Understanding whether a series converges (approaches a finite limit) or diverges is fundamental in calculus. Convergence tests help determine this behavior without explicitly finding the sum.
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Convergence of an Infinite Series

Simplifying Rational Expressions

Before applying convergence tests, it is often helpful to simplify the general term of the series. Factoring the denominator or numerator can reveal cancellations or simpler forms, making it easier to identify the dominant behavior of terms as k approaches infinity.
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Simplifying Trig Expressions

Convergence Tests for Series

Common tests include the Comparison Test, Limit Comparison Test, and the Divergence Test. Choosing the right test depends on the form of the series terms; for rational functions, comparing to a simpler p-series or using the Limit Comparison Test is often effective.
Recommended video:
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Choosing a Convergence Test
Related Practice
Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0

Textbook Question

Define the remainder of an infinite series.

Textbook Question

13–52. Limits of sequences

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


{tan⁻¹(10n⁄(10n + 4))}  

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Textbook Question

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.


{(−0.7)ⁿ}

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Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


21.∑ (k = 0 to ∞) (1/4)ᵏ

Textbook Question

Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k. 

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