Domanda del libro di testo
∑ (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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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 10.7.36b
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 ]
Guida verificata passo dopo passo1
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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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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07:36
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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Percorso guidato
06:52Convergence 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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Percorso guidato
07:45Limit Comparison Test
Pratica correlata
Domanda del libro di testo
b. From Example 5, Section 10.2, show that
S = 1 + ∑(from n=1 to ∞) [1 / (n²(n + 1))].
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Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
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) ]
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