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 ∞) sin (1/n)
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 10.8.37
In Exercises 35–40, find the first three nonzero terms of the Maclaurin series for each function.
f(x) = (sin x) ln(1 + x)
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 three nonzero terms of the Maclaurin series for \(f(x) = (\sin x) \ln(1 + x)\).
Write down the Maclaurin series expansions for each component function separately:
\(\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\)
and
\(\ln(1+x) = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\).
Multiply the two series together term-by-term, keeping track of powers of \(x\). Specifically, multiply each term of \(\sin x\) by each term of \(\ln(1+x)\) and combine like powers of \(x\).
Identify and collect the first three nonzero terms from the resulting series after multiplication. This involves adding coefficients of like powers of \(x\) and ignoring terms that are zero.
Write the resulting expression as the sum of these first three nonzero terms, which will be the beginning of the Maclaurin series for \(f(x) = (\sin x) \ln(1 + x)\).
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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 of the function near x = 0.
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Convergence of Taylor & Maclaurin Series
Series Expansion of Elementary Functions
Common functions like sin(x) and ln(1 + x) have known Maclaurin series expansions. Understanding these expansions helps in combining or manipulating series to find the series of more complex functions.
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Multiplication of Power Series
To find the Maclaurin series of a product like (sin x)·ln(1 + x), multiply the individual series term-by-term and combine like powers of x. This process requires careful organization to identify the first nonzero terms.
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Related Practice
Textbook Question
Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.
(tan⁻¹x)²
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 81–86, find the values of x for which the given geometric series converges.
Also, find the sum of the series (as a function of x) for those values of x.
∑ (from n = 0 to ∞) [ (−1/2)ⁿ (x − 3)ⁿ ]
Textbook Question
Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)