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 ∞) [(-1)ⁿ n² e⁻ⁿ]
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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 10.7.18
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 = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]
Guida verificata passo dopo passo1
Identify the general term of the series: \(a_n = \frac{n x^n}{4^n (n^2 + 1)}\).
Apply the Ratio Test to find the radius of convergence. Compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\):
\[L = \lim_{n \to \infty} \left| \frac{(n+1) x^{n+1}}{4^{n+1} ((n+1)^2 + 1)} \cdot \frac{4^n (n^2 + 1)}{n x^n} \right| = \lim_{n \to \infty} \left| \frac{n+1}{n} \cdot \frac{n^2 + 1}{(n+1)^2 + 1} \cdot \frac{|x|}{4} \right|.\]
Simplify the limit by analyzing the behavior of the rational expressions as \(n \to \infty\), which will approach 1, so \(L = \frac{|x|}{4}\).
Set the condition for convergence from the Ratio Test: \(L < 1\), which gives \(\frac{|x|}{4} < 1\), or \(|x| < 4\). This means the radius of convergence is \(R = 4\) and the interval of convergence is initially \((-4, 4)\).
Check the endpoints \(x = -4\) and \(x = 4\) by substituting into the original series and testing for convergence (using appropriate tests such as the Alternating Series Test or p-series test) to determine if the series converges absolutely, conditionally, or diverges at these points.
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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. Finding these involves applying tests like the Ratio or Root Test to determine where the series converges absolutely.
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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, ensuring the original series converges regardless of term signs. Absolute convergence implies stronger convergence and often simplifies analysis of power series.
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Percorso guidato
07:51Choosing a Convergence Test
Conditional Convergence
Conditional convergence occurs when a series converges, but not absolutely; that is, the series ∑a_n converges while ∑|a_n| diverges. This typically happens at the endpoints of the interval of convergence and requires careful testing, such as the Alternating Series Test, to confirm.
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Percorso guidato
07:51Choosing a Convergence Test
Pratica correlata
Domanda del libro di testo
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
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Domanda del libro di testo
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
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Domanda del libro di testo
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 / (n³ 3ⁿ) ]
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Domanda del libro di testo
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]
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Domanda del libro di testo
Use series to evaluate the limits in Exercises 29–40.
37. lim (x → 0) ln(1 + x²) / (1 - cos(x))
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