Domanda del libro di testo
Error Estimates
The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.
30
views
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 10.5.60
Convergence or Divergence
Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]
Guida verificata passo dopo passo1
Identify the given series: \( \sum_{n=1}^{\infty} \frac{(-1)^n (n!)^n}{n^{n^2}} \). We want to determine whether this series converges or diverges.
Since the series involves factorials and powers raised to powers, consider using the Root Test, which is effective for series with terms raised to the \( n \)-th power. The Root Test uses the limit \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \), where \( a_n = \frac{(-1)^n (n!)^n}{n^{n^2}} \).
Calculate \( \sqrt[n]{|a_n|} = \sqrt[n]{\frac{(n!)^n}{n^{n^2}}} = \frac{n!}{n^{n}} \) because \( \sqrt[n]{(n!)^n} = n! \) and \( \sqrt[n]{n^{n^2}} = n^{n} \).
Analyze the behavior of \( \frac{n!}{n^n} \) as \( n \to \infty \). Use Stirling's approximation for factorials: \( n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \). Substitute this approximation to understand the limit.
Evaluate the limit \( L = \lim_{n \to \infty} \frac{n!}{n^n} \approx \lim_{n \to \infty} \frac{\sqrt{2 \pi n} \left( \frac{n}{e} \right)^n}{n^n} = \lim_{n \to \infty} \sqrt{2 \pi n} \left( \frac{1}{e} \right)^n \). Since \( \left( \frac{1}{e} \right)^n \) tends to zero exponentially, \( L = 0 \). According to the Root Test, if \( L < 1 \), the series converges absolutely.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Convergence and Divergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely; otherwise, it diverges. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to assess whether the series sums to a finite value.
Video consigliato:
Percorso guidato
06:52
Convergence of an Infinite Series
Ratio Test
The Ratio Test evaluates the limit of the absolute value of the ratio of consecutive terms in a series. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. This test is especially useful for series involving factorials and exponential terms.
Video consigliato:
Percorso guidato
05:35Ratio Test
Factorials and Exponential Growth
Factorials (n!) grow faster than exponential functions as n increases, which significantly affects the behavior of series terms. Understanding how factorials compare to powers and exponentials helps in applying convergence tests and predicting whether terms shrink quickly enough for the series to converge.
Video consigliato:
5:22Factorials
Pratica correlata
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))]
31
views
Domanda del libro di testo
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 2 + (0.1)ⁿ
18
views
Domanda del libro di testo
In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]
37
views
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 / (2√n + ³√n)
31
views
Domanda del libro di testo
Using the Ratio Test
In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.
∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]
6
views