Domanda del libro di testo
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 0 to ∞) e^(−2n)
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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 10.7.10
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]
Guida verificata passo dopo passo1
Identify the given power series: \(\sum_{n=1}^{\infty} \frac{(x - 1)^n}{\sqrt{n}}\) and recognize that it is centered at \(x = 1\).
To find the radius of convergence, apply the Root Test or Ratio Test. Here, the Root Test is convenient: consider \(\lim_{n \to \infty} \sqrt[n]{\left| \frac{(x - 1)^n}{\sqrt{n}} \right|} = \lim_{n \to \infty} \frac{|x - 1|}{n^{1/(2n)}}\).
Since \(\lim_{n \to \infty} n^{1/(2n)} = 1\), the limit simplifies to \(|x - 1|\). For convergence, this limit must be less than 1, so the radius of convergence is \(R = 1\) and the interval of convergence is initially \(1 - 1 < x < 1 + 1\), or \((0, 2)\).
Next, check convergence at the endpoints \(x = 0\) and \(x = 2\) by substituting into the series: at \(x=0\), the series becomes \(\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\); at \(x=2\), it becomes \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\). Analyze these using the Alternating Series Test and p-series test respectively.
Determine absolute convergence by testing \(\sum_{n=1}^{\infty} \left| \frac{(x - 1)^n}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{|x - 1|^n}{\sqrt{n}}\). This series converges absolutely when \(|x - 1| < 1\). For conditional convergence, check if the series converges at endpoints where absolute convergence fails.
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Radius and Interval of Convergence
The radius of convergence is the distance from the center of a power series within which the series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. It is found using tests like the Ratio or Root Test applied to the general term.
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07:36
Radius of Convergence
Absolute Convergence
A series converges absolutely if the series of absolute values converges. This means ∑|a_n| converges, which guarantees convergence regardless of term signs. Absolute convergence implies regular convergence and is often easier to test using comparison or root tests.
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Percorso guidato
07:51Choosing a Convergence Test
Conditional Convergence
Conditional convergence occurs when a series converges, but does not converge absolutely. This means the series ∑a_n converges, but ∑|a_n| diverges. It often happens at the endpoints of the interval of convergence and requires tests like the Alternating Series Test.
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Percorso guidato
07:51Choosing a Convergence Test
Pratica correlata
Domanda del libro di testo
Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.
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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 = 0 to ∞) [((n + 1) / (n + 2))ⁿ]
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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 ∞) (n / (3n + 1))ⁿ
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Domanda del libro di testo
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))
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Domanda del libro di testo
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4
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