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
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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))}
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas 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 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.
Recommended video:
Guided course
8:22
Introduction to Sequences
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.
Recommended video:
6:04Intro to Rational Functions
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.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
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.
∑ (k = 1 to ∞) (5 / 6)⁻ᵏ
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 = 10 to ∞) 1 / (k − 9)⁵
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 = 3 to ∞) (2k²) / (k² − k − 2)
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}
1
views
Textbook Question
Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k.
1
views