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)
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 10.6.72
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))
Guida verificata passo dopo passo1
First, analyze the general term of the series: \(a_n = \tan\left(n^{\frac{1}{n}}\right)\). To determine convergence or divergence, we need to understand the behavior of \(a_n\) as \(n\) approaches infinity.
Evaluate the limit of the inside of the tangent function: \(\lim_{n \to \infty} n^{\frac{1}{n}}\). Recall that \(n^{\frac{1}{n}} = e^{\frac{\ln n}{n}}\). Since \(\frac{\ln n}{n} \to 0\) as \(n \to \infty\), this limit approaches \(e^0 = 1\).
Use the limit found to find \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \tan\left(n^{\frac{1}{n}}\right) = \tan(1)\). Since \(\tan(1)\) is a finite nonzero number, the terms \(a_n\) do not approach zero.
Recall the Divergence Test (also called the nth-term test for divergence): if \(\lim_{n \to \infty} a_n \neq 0\), then the series \(\sum a_n\) diverges. Since \(\lim_{n \to \infty} a_n = \tan(1) \neq 0\), the series diverges.
Therefore, conclude that the series \(\sum_{n=1}^\infty \tan\left(n^{\frac{1}{n}}\right)\) diverges by the Divergence Test.
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Behavior of the General Term in a Series
To determine convergence, first analyze the limit of the general term as n approaches infinity. If the term does not approach zero, the series diverges by the Test for Divergence. Understanding the behavior of tan(n^(1/n)) as n grows large is crucial here.
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Percorso guidato
06:00
Geometric Series
Limit Comparison and Standard Convergence Tests
Various tests like the Comparison Test, Limit Comparison Test, or the Divergence Test help determine series convergence. Choosing an appropriate test depends on the form of the terms; for example, comparing tan(n^(1/n)) to a simpler function can clarify convergence behavior.
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Percorso guidato
07:45Limit Comparison Test
Properties and Limits of the Function n^(1/n)
The expression n^(1/n) approaches 1 as n approaches infinity. Understanding this limit helps simplify tan(n^(1/n)) to approximately tan(1) for large n, which informs whether the terms tend to zero or not, a key step in analyzing series convergence.
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Percorso guidato
06:21Properties of Functions
Pratica correlata
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
Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.
(tan⁻¹x)²
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)
Domanda del libro di testo
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 ]