Domanda del libro di testo
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) ]
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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 10.5.66a
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]
Guida verificata passo dopo passo1
Identify the general term of the series: \( a_n = \frac{b_{n+1} + b_n}{n 4^n} \). Since \( b_n \) converges to \( \frac{1}{3} \), both \( b_n \) and \( b_{n+1} \) approach \( \frac{1}{3} \) as \( n \to \infty \).
Use the limit of \( b_n \) to approximate the behavior of the numerator for large \( n \): \( b_{n+1} + b_n \to \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \). This helps simplify the term for large \( n \).
Focus on the denominator \( n 4^n \), which grows very rapidly due to the exponential term \( 4^n \). This suggests the terms \( a_n \) decrease quickly.
Apply the Comparison Test or Limit Comparison Test by comparing \( a_n \) to a known convergent series, such as \( \frac{1}{4^n} \), since \( \frac{b_{n+1} + b_n}{n 4^n} \) behaves similarly to \( \frac{constant}{n 4^n} \).
Conclude about convergence: since \( \sum \frac{1}{4^n} \) converges (geometric series with ratio \( \frac{1}{4} < 1 \)) and \( \frac{1}{n} \) grows slowly, the given series converges by comparison.
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Convergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence involves analyzing the behavior of terms as n approaches infinity, ensuring the sum does not diverge to infinity or oscillate indefinitely.
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Percorso guidato
06:52
Convergence of an Infinite Series
Limit of a Sequence
The limit of a sequence is the value that its terms approach as n becomes very large. Knowing that bₙ converges to 1/3 helps approximate terms in the series for large n, which is crucial for applying convergence tests effectively.
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8:22Introduction to Sequences
Comparison and Root/Ratio Tests for Series
These tests help determine series convergence by comparing terms to known convergent series or analyzing the nth root or ratio of successive terms. Given the presence of 4ⁿ in the denominator, the ratio test is particularly useful to assess the exponential decay of terms.
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Percorso guidato
07:15Root Test
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)]
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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
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Domanda del libro di testo
Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)
f(x) = 1 / √(1 − x²)
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Domanda del libro di testo
The series
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + ⋯
converges to eˣ for all x.
a. Find a series for (d/dx)eˣ. Do you get the series for eˣ? Explain your answer.
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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.
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
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