Domanda del libro di testo
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
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TOC.tableOfContents
- 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
Capitolo 10, Problema 10.9.33
Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.
(tan⁻¹x)²
Guida verificata passo dopo passo1
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 \((\tan^{-1} x)^2\).
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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.
Video consigliato:
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.
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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.
Video consigliato:
Percorso guidato
06:00Geometric Series
Pratica correlata
Domanda del libro di testo
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)
Domanda del libro di testo
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]
Domanda del libro di testo
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))
Domanda del libro di testo
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)!]
Domanda del libro di testo
In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.
f(x) = (sin x) ln(1 + x)