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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 10.4.53
Chapter 10, Problem 10.4.53

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 ∞) k / √(k² + 1)

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First, write down the general term of the series: \(a_k = \frac{k}{\sqrt{k^2 + 1}}\).
To analyze convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Simplify the expression inside the square root to understand the dominant terms.
Divide numerator and denominator by \(k\) to rewrite \(a_k\) as \(\frac{k}{\sqrt{k^2 + 1}} = \frac{k}{k \sqrt{1 + \frac{1}{k^2}}} = \frac{1}{\sqrt{1 + \frac{1}{k^2}}}\).
Evaluate the limit \(\lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{1}{\sqrt{1 + \frac{1}{k^2}}}\) to determine if the terms approach zero, which is necessary for convergence.
Since the terms do not approach zero, conclude that the series diverges by the Test for Divergence (also known as the nth-term test).

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Key Concepts

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

Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.
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Convergence of an Infinite Series

Limit Comparison Test

The Limit Comparison Test compares a given series with a known benchmark series by examining the limit of their term ratios. If the limit is a positive finite number, both series either converge or diverge together. This test is useful when terms resemble a simpler series.
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Limit Comparison Test

Behavior of Terms and nth-Term Test for Divergence

The nth-term test states that if the limit of the terms of a series does not approach zero, the series must diverge. Analyzing the behavior of the general term, especially for large n, helps quickly identify divergence without further tests.
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Divergence Test (nth Term Test)
Related Practice
Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))

Textbook Question

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).


49.0.037̅ = 0.037037…

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.


57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))

Textbook Question

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

∑ (k = 1 to ∞) cos(k) / k³

Textbook Question

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


27.1 + 1.01 + 1.01² + 1.01³ + ⋯

Textbook Question

13–52. Limits of sequences

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


{(3n³ − 1)⁄(2n³ + 1)}

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