Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (1 + cos n) / n²
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.9.33
Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.
(tan⁻¹x)²
Verified step by step guidance1
Recall that the Maclaurin series is the Taylor series expansion of a function about \(x=0\). We want to find the first four nonzero terms of the Maclaurin series for the function \(f(x) = (\tan^{-1} x)^2\).
Start by writing the Maclaurin series for \(\tan^{-1} x\). The known expansion is:
\[
\tan^{-1} x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots
\]
Since \(f(x) = (\tan^{-1} x)^2\), square the series for \(\tan^{-1} x\) up to the terms that will produce the first four nonzero terms in \(f(x)\). This means considering terms up to at least \(x^7\) in \(\tan^{-1} x\) to get terms up to \(x^8\) in \(f(x)\).
Perform the multiplication:
\[
\left(x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \right)^2
\]
Multiply out the terms carefully, combining like powers of \(x\), and keep track of the coefficients for each power.
Collect the terms by powers of \(x\) and identify the first four nonzero terms in the resulting series. These will be the first four nonzero terms of the Maclaurin series for \((\tan^{-1} x)^2\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Maclaurin Series
A Maclaurin series is a Taylor series expansion of a function about zero. It expresses the function as an infinite sum of terms involving derivatives evaluated at zero, allowing approximation near x = 0. Understanding how to find and use these series is essential for expanding functions like (tan⁻¹x)².
Recommended video:
08:26
Convergence of Taylor & Maclaurin Series
Series Expansion of Inverse Trigonometric Functions
The inverse tangent function, tan⁻¹x, has a known Maclaurin series expansion involving odd powers of x with alternating signs. Recognizing and using this series is crucial for finding the expansion of (tan⁻¹x)² by substituting and manipulating the series terms.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Multiplication and Collection of Series Terms
To find the series for (tan⁻¹x)², one must square the Maclaurin series of tan⁻¹x and combine like powers of x. This involves multiplying series term-by-term and carefully collecting coefficients of each power to identify the first four nonzero terms.
Recommended video:
06:00Geometric Series
Related Practice
Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) 1 / (1 + 2² + 3² + ⋯ + n²)
Textbook Question
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=3 to ∞) [2n² / n²ⁿ]
Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 1 to ∞) tan(n^(1/n))
Textbook Question
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [n! / (2n + 1)!]
Textbook Question
In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.
f(x) = (sin x) ln(1 + x)