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 10.2.63

Chapter 10, Problem 10.2.63

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{tan⁻¹n / n}

Verified step by step guidance

Identify the given sequence as \( a_n = \frac{\tan^{-1}(n)}{n} \), where \( \tan^{-1}(n) \) is the inverse tangent (arctangent) function.

Recall the behavior of the arctangent function as \( n \to \infty \): \( \tan^{-1}(n) \) approaches \( \frac{\pi}{2} \) because the arctangent of very large positive numbers tends to \( \frac{\pi}{2} \).

Rewrite the limit expression using this information: \( \lim_{n \to \infty} \frac{\tan^{-1}(n)}{n} = \lim_{n \to \infty} \frac{\frac{\pi}{2}}{n} \) approximately for large \( n \).

Since the denominator \( n \) grows without bound and the numerator approaches a constant \( \frac{\pi}{2} \), analyze the limit of a constant divided by an infinitely large number.

Conclude that the limit of the sequence is \( 0 \) because dividing a finite constant by an infinitely large number tends to zero.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 describes the value that the terms of the sequence approach as the index goes to infinity. Understanding how to evaluate limits helps determine whether a sequence converges to a finite number or diverges.

Behavior of the Arctangent Function

The arctangent function, tan⁻¹(x), is continuous and bounded, approaching π/2 as x approaches infinity. Knowing this asymptotic behavior is crucial for analyzing sequences involving tan⁻¹(n) as n grows large.

Growth Rates and Dominant Terms

Comparing the growth rates of numerator and denominator helps determine the limit of a sequence. Since n grows without bound and tan⁻¹(n) approaches a constant, understanding which term dominates is key to finding the sequence's limit.

Recommended video:

04:16

Intro To Related Rates

Related Practice

Textbook Question

55–70. More sequences

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

aₙ = (−1)ⁿ ⁿ√n

Textbook Question

13–52. Limits of sequences

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

{(3ⁿ⁺¹ + 3)⁄3ⁿ}

Textbook Question

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

∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)

Textbook Question

13–52. Limits of sequences

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

{(1 + (2 / n))ⁿ}

views

Textbook Question

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 = 1 to ∞) (k² / (k⁴ + k³ + 1))

Textbook Question

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.

2 / 4² + 2 / 5² + 2 / 6² + ⋯

55–70. More sequencesFind the limit of the following sequences or determine that the sequence diverges.{tan⁻¹n / n}

Verified video answer for a similar problem:

Key Concepts

Limits of Sequences

Behavior of the Arctangent Function

Growth Rates and Dominant Terms

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{tan⁻¹n / n}