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?
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 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
Chapter 10, Problem 10.7.28b
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 ]
Verified step by step guidance1
Identify the given power series: \(\sum_{n=0}^{\infty} (-2)^n (n+1) (x-1)^n\).
To determine absolute convergence, consider the series formed by the absolute values of the terms: \(\sum_{n=0}^{\infty} |(-2)^n (n+1) (x-1)^n| = \sum_{n=0}^{\infty} (n+1) |2|^n |x-1|^n = \sum_{n=0}^{\infty} (n+1) (2|x-1|)^n\).
Recognize that this is a power series in terms of \(r = 2|x-1|\) with general term \((n+1) r^n\). To analyze convergence, use the root or ratio test, or recall the known result for the series \(\sum (n+1) r^n\).
Recall that the series \(\sum_{n=0}^{\infty} (n+1) r^n\) converges if and only if \(|r| < 1\). Therefore, set up the inequality \(2|x-1| < 1\) to find the interval where the series converges absolutely.
Solve the inequality \(2|x-1| < 1\) to find the values of \(x\) for which the original series converges absolutely. This will give the interval of absolute convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radius and Interval of Convergence
The radius of convergence defines the distance from the center point within which a power series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. Finding this interval involves applying tests like the Ratio or Root Test to the general term.
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07:36
Radius of Convergence
Absolute Convergence
A series converges absolutely if the series of absolute values converges. This is a stronger form of convergence ensuring the original series converges regardless of term signs. Testing absolute convergence often simplifies analysis by removing alternating signs.
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07:51Choosing a Convergence Test
Ratio Test for Series Convergence
The Ratio Test evaluates the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is especially useful for power series involving factorials or exponential terms.
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05:35Ratio Test
Related Practice
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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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
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) = ln(cos x)
Textbook Question
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)
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 ]