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 10.7.40
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ n! xⁿ / nⁿ ]
Guida verificata passo dopo passo1
Identify the general term of the series as \(a_n = \frac{n! x^n}{n^n}\).
Recall that the radius of convergence \(R\) can be found using the Root Test or the Ratio Test. Here, the Ratio Test is convenient, which involves evaluating \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
Write the ratio \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(n+1)! x^{n+1} / (n+1)^{n+1}}{n! x^n / n^n} \right| = |x| \cdot \frac{(n+1)!}{n!} \cdot \frac{n^n}{(n+1)^{n+1}}\).
Simplify the factorial and powers: \(\frac{(n+1)!}{n!} = n+1\), so the ratio becomes \(|x| (n+1) \frac{n^n}{(n+1)^{n+1}}\).
Rewrite \(\frac{n^n}{(n+1)^{n+1}}\) as \(\frac{n^n}{(n+1)^n (n+1)} = \frac{1}{n+1} \left( \frac{n}{n+1} \right)^n\), then simplify the entire expression and take the limit as \(n \to \infty\) to find the radius of convergence \(R\).
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Radius of Convergence
The radius of convergence of a power series is the distance from the center of the series within which the series converges absolutely. It determines the interval on the x-axis where the series sums to a finite value. Finding this radius helps understand the domain of validity for the series representation.
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07:36
Radius of Convergence
Ratio Test for Series Convergence
The ratio test is a method to determine the convergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. For power series, it is commonly used to find the radius of convergence by setting the limit of |a_{n+1}/a_n| equal to 1/|x|.
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Percorso guidato
05:35Ratio Test
Stirling's Approximation
Stirling's approximation provides an estimate for factorials, especially useful for large n, stating n! ≈ √(2πn) (n/e)^n. This approximation simplifies factorial expressions in series terms, making it easier to analyze limits and apply convergence tests.
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Percorso guidato
04:57Determining Error and Relative Error
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 − n) / n2ⁿ
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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
Use series to evaluate the limits in Exercises 29–40.
37. lim (x → 0) ln(1 + x²) / (1 - cos(x))
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Domanda del libro di testo
Applying the Integral Test
Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.
∑ (from n = 1 to ∞) 1 / n⁰·²
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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 = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]
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