Textbook Question
Assume that bₙ is a sequence of positive numbers converging to 4/5. Determine if the following series converge or diverge.
b. ∑ (from n = 1 to ∞) (5/4)ⁿ (bₙ)
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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 10.4.74b
b. From Example 5, Section 10.2, show that
S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].
Verified step by step guidance1
Recall the series expression from Example 5, Section 10.2, which involves the sum of terms related to \( \frac{1}{n^2 (n+1)} \). Our goal is to express \( S \) as \( 1 + \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \).
Start by writing the general term of the series: \( \frac{1}{n^2 (n+1)} \). To simplify or analyze this term, consider using partial fraction decomposition to break it into simpler fractions that are easier to sum.
Set up the partial fraction decomposition for \( \frac{1}{n^2 (n+1)} \) as \( \frac{A}{n} + \frac{B}{n^2} + \frac{C}{n+1} \), and solve for constants \( A, B, C \) by multiplying both sides by \( n^2 (n+1) \) and equating coefficients.
Once the partial fractions are found, rewrite the sum \( \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \) as the sum of simpler series involving \( \sum \frac{1}{n} \), \( \sum \frac{1}{n^2} \), and \( \sum \frac{1}{n+1} \).
Recognize that the term \( 1 \) outside the summation corresponds to the initial term or a boundary condition from the original series, completing the expression \( S = 1 + \sum_{n=1}^\infty \frac{1}{n^2 (n+1)} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Summation Notation
An infinite series is the sum of infinitely many terms, often expressed using summation notation (∑). Understanding how to interpret and manipulate these sums is essential for evaluating or transforming series expressions.
Recommended video:
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Convergence of an Infinite Series
Partial Fraction Decomposition
Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique is useful for simplifying terms like 1/(n²(n+1)) to facilitate summation or integration.
Recommended video:
10:07Partial Fraction Decomposition: Distinct Linear Factors
Telescoping Series
A telescoping series is one where many terms cancel out when the series is expanded, leaving only a few terms to sum. Recognizing and applying telescoping can simplify the evaluation of series like the one given.
Recommended video:
06:00Geometric Series
Related Practice
Textbook Question
A sequence of rational numbers is described as follows:
1/1,3/2,7/5,17/12,…,a/b,(a + 2b)/(a + b),…
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xₙ and yₙ be, respectively, the numerator and the denominator of the nᵗʰ fraction rₙ = xₙ / yₙ.
b. The fractions rₙ = xₙ / yₙ approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that rₙ² − 2 = ±(1 / yₙ)² and that yₙ is not less than n.)
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Textbook Question
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]
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
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
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 ]