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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 10.2.17
Chapter 10, Problem 10.2.17

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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1
Identify the general term of the sequence: \(a_n = \tan^{-1}\left(\frac{10n}{10n + 4}\right)\).
Analyze the behavior of the argument inside the inverse tangent function as \(n\) approaches infinity: consider the limit of \(\frac{10n}{10n + 4}\) as \(n \to \infty\).
Simplify the fraction \(\frac{10n}{10n + 4}\) by dividing numerator and denominator by \(n\), yielding \(\frac{10}{10 + \frac{4}{n}}\).
Evaluate the limit of the simplified fraction as \(n \to \infty\), noting that \(\frac{4}{n} \to 0\), so the fraction approaches \(\frac{10}{10} = 1\).
Use the continuity of the \(\tan^{-1}(x)\) function to conclude that the limit of the sequence is \(\tan^{-1}(1)\), which you can recognize or leave in this form.

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Key Concepts

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

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
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Behavior of Rational Functions as n → ∞

For sequences involving rational expressions like 10n/(10n + 4), as n becomes very large, lower-order terms become negligible. The expression approaches the ratio of the leading coefficients, which helps simplify the limit evaluation.
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Continuity and Limits of Inverse Trigonometric Functions

Inverse trigonometric functions like arctan are continuous, so the limit of arctan(f(n)) as n → ∞ equals arctan of the limit of f(n), provided the limit of f(n) exists. This property allows evaluating limits inside the function.
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Related Practice
Textbook Question

For what values of r does the sequence {rⁿ} converge? Diverge?

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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 k³ / √(k⁸ + 1)

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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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Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)

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Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((k / (k + 1)) × 2k²)

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Textbook Question

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


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)

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