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]
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 10.5.37
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)!]
Guida verificata passo dopo passo1
First, identify the general term of the series: \(a_n = \frac{n!}{(2n + 1)!}\).
Since the terms involve factorials, consider using the Ratio Test, which is effective for series with factorial expressions.
Write the Ratio Test limit: \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{(n+1)!}{(2(n+1) + 1)!} \cdot \frac{(2n + 1)!}{n!}\).
Simplify the factorial expressions inside the limit by expanding \((n+1)! = (n+1) \cdot n!\) and \((2(n+1) + 1)! = (2n + 3)! = (2n + 3)(2n + 2)(2n + 1)!\).
After simplification, evaluate the limit \(L\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.
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:
5mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Factorial Growth and Comparison
Factorials grow very rapidly compared to polynomial or exponential functions. Understanding how n! compares to (2n+1)! is crucial for analyzing the behavior of the terms in the series and determining if they approach zero fast enough for convergence.
Video consigliato:
5:22
Factorials
Ratio Test for Series Convergence
The Ratio Test involves taking the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. This test is especially useful for series involving factorials.
Video consigliato:
Percorso guidato
05:35Ratio Test
Definition of Convergence and Divergence of Infinite Series
A series converges if the sequence of its partial sums approaches a finite limit; otherwise, it diverges. Recognizing this fundamental definition helps in applying tests and interpreting their results correctly.
Video consigliato:
Percorso guidato
06:52Convergence of an Infinite Series
Pratica correlata
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
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))
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 ]
Domanda del libro di testo
Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.
sin x – x + (x³ / 3!)
49
views