Textbook Question
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]
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.3.62a
(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)]
Verified step by step guidance1
Recall the result from Exercise 61, which likely involved the Integral Test for series convergence, especially for series of the form \( \sum \frac{1}{n (\ln n)^p} \). The Integral Test states that if \( f(x) = \frac{1}{x \ln x} \) is positive, continuous, and decreasing for \( x \geq 2 \), then the convergence of the series \( \sum_{n=2}^\infty \frac{1}{n \ln n} \) is determined by the convergence of the integral \( \int_2^\infty \frac{1}{x \ln x} \, dx \).
Set up the integral to apply the Integral Test: \[ \int_2^\infty \frac{1}{x \ln x} \, dx. \] This integral will help us determine if the series converges or diverges.
Use the substitution method to evaluate the integral: let \( u = \ln x \), so that \( du = \frac{1}{x} dx \). This transforms the integral into \[ \int_{\ln 2}^\infty \frac{1}{u} \, du. \]
Recognize that \( \int \frac{1}{u} \, du = \ln |u| + C \). Therefore, the integral becomes \[ \lim_{t \to \infty} \int_{\ln 2}^t \frac{1}{u} \, du = \lim_{t \to \infty} (\ln t - \ln (\ln 2)) = \infty. \]
Since the integral diverges to infinity, by the Integral Test, the series \( \sum_{n=2}^\infty \frac{1}{n \ln n} \) also diverges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Comparison Test for Series
The Comparison Test helps determine convergence or divergence by comparing a given series to another series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than those of a divergent series, it diverges.
Recommended video:
09:25
Direct Comparison Test
Integral Test
The Integral Test relates the convergence of a series to the convergence of an improper integral of a related function. If the integral of f(x) from some point to infinity converges, then the series ∑ f(n) converges; if the integral diverges, so does the series.
Recommended video:
07:25Integral Test
Behavior of the Harmonic Series and Logarithmic Modifications
The harmonic series ∑ 1/n diverges, but adding logarithmic terms in the denominator, such as ∑ 1/(n ln n), affects convergence. Understanding how slowly the logarithm grows is key to analyzing whether such series converge or diverge.
Recommended video:
04:30P-Series and Harmonic Series
Related Practice
Textbook Question
∑ (from n=1 to ∞) (1 / √(n + 1)) diverges
b. What should n be in order that the partial sum sₙ = ∑ (from i=1 to n) (1 / √(i + 1)) satisfies sₙ > 1000?
Textbook Question
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?
Textbook Question
Use the Cauchy condensation test from Exercise 59 to show that:
b. ∑ (from n=1 to ∞) [1 / nᵖ] converges if p > 1 and diverges if p ≤ 1.
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Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence.
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]
Textbook Question
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]