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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.7.36b
Chapter 10, Problem 10.7.36b

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 ]

Verified step by step guidance
1
Identify the general term of the series: \[a_n = (\sqrt{n+1} - \sqrt{n})(x - 3)^n\].
To determine absolute convergence, consider the series of absolute values: \[\sum_{n=1}^\infty |\sqrt{n+1} - \sqrt{n}| \cdot |x - 3|^n\].
Simplify the term \[|\sqrt{n+1} - \sqrt{n}|\] by rationalizing the difference: \[\sqrt{n+1} - \sqrt{n} = \frac{(n+1) - n}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}}\].
Analyze the behavior of \[a_n = \frac{1}{\sqrt{n+1} + \sqrt{n}} |x - 3|^n\] for large \[n\]. Since \[\frac{1}{\sqrt{n+1} + \sqrt{n}}\] behaves like \[\frac{1}{2\sqrt{n}}\], the term behaves roughly like \[\frac{|x - 3|^n}{\sqrt{n}}\].
Use the root test or ratio test on the simplified term to find the radius of convergence. For absolute convergence, the series converges when \[|x - 3| < 1\]. Then check the endpoints \[x = 2\] and \[x = 4\] separately to determine if the series converges absolutely there.

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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. Determining this interval involves applying tests like the Ratio or Root Test to the series terms.
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Radius of Convergence

Absolute Convergence of Series

A series converges absolutely if the series of absolute values of its terms converges. Absolute convergence implies convergence, and it is often easier to test using comparison or root tests. For power series, absolute convergence is checked by considering |x - center| within the radius.
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Convergence of an Infinite Series

Behavior of the General Term and Limit Comparison

Analyzing the general term, especially expressions like (√(n+1) − √n), helps simplify and understand the series' behavior. Using limit comparison or asymptotic approximations (e.g., √(n+1) − √n ≈ 1/(2√n)) aids in applying convergence tests effectively.
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Limit Comparison Test
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?

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Textbook Question

b. From Example 5, Section 10.2, show that

S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].

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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)

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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 = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]

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