Domanda del libro di testo
Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.
a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]
17
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 10.7.32a
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence.
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
Guida verificata passo dopo passo1
Identify the general term of the series: \( a_n = \frac{(3x + 1)^{n+1}}{2n + 2} \).
To find the radius of convergence, apply the Root Test or Ratio Test. Here, the Ratio Test is convenient. Consider \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
Compute the ratio \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(3x + 1)^{n+2}}{2(n+1) + 2} \cdot \frac{2n + 2}{(3x + 1)^{n+1}} \right| = \left| (3x + 1) \right| \cdot \frac{2n + 2}{2n + 4} \).
Take the limit as \( n \to \infty \) of the ratio: \( \lim_{n \to \infty} \left| (3x + 1) \right| \cdot \frac{2n + 2}{2n + 4} = \left| 3x + 1 \right| \cdot 1 = \left| 3x + 1 \right| \).
Set the limit less than 1 for convergence: \( \left| 3x + 1 \right| < 1 \). Solve this inequality to find the interval of convergence, then check the endpoints \( x \) values by substituting back into the original series to determine if the series converges at those points.
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:
12mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Radius of Convergence
The radius of convergence is the distance from the center of a power series within which the series converges absolutely. It can be found using the Ratio Test or Root Test on the general term of the series. This radius defines an interval on the x-axis where the series behaves well and sums to a finite value.
Video consigliato:
07:36
Radius of Convergence
Interval of Convergence
The interval of convergence is the set of all x-values for which the power series converges. It is centered at the series’ expansion point and extends to the radius of convergence on both sides. Endpoints must be tested separately to determine if the series converges or diverges there.
Video consigliato:
08:44Interval of Convergence
Applying the Ratio Test to Power Series
The Ratio Test compares the limit of the absolute value of consecutive terms to determine convergence. For power series, it helps find the radius of convergence by simplifying the ratio of terms involving x. If the limit is less than one, the series converges; if greater, it diverges.
Video consigliato:
05:58Intro to Power Series
Pratica correlata
Domanda del libro di testo
(Continuation of Exercise 61.) Use the result in Exercise 61 to determine which of the following series converge and which diverge. Support your answer in each case.
a. ∑ (from n=2 to ∞) [1 / (n ln n)]
5
views
Domanda del libro di testo
The series
sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯
converges to sec x for −π/2 < x < π/2.
a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?
30
views
Domanda del libro di testo
Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
a. Converges absolutely for x = 1
19
views
Domanda del libro di testo
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (-4)ⁿ/n!
24
views
Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence.
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
18
views