Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]
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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 10.7.36c
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]
Guida verificata passo dopo passo1
Identify the general term of the series: \[a_n = (\sqrt{n+1} - \sqrt{n})(x - 3)^n\].
Simplify the coefficient \[\sqrt{n+1} - \sqrt{n}\] by rationalizing the numerator: multiply numerator and denominator by \[\sqrt{n+1} + \sqrt{n}\] to get \[\frac{(n+1) - n}{\sqrt{n+1} + \sqrt{n}} = \frac{1}{\sqrt{n+1} + \sqrt{n}}\].
Rewrite the general term as \[a_n = \frac{(x - 3)^n}{\sqrt{n+1} + \sqrt{n}}\] and note that for large \[n\], \[\sqrt{n+1} + \sqrt{n} \approx 2\sqrt{n}\], so \[a_n \sim \frac{(x - 3)^n}{2\sqrt{n}}\].
Use the Root Test or Ratio Test to find the radius of convergence by focusing on the \[|x - 3|\] term, since the denominator grows like \[\sqrt{n}\] which affects convergence but not the radius directly.
Check convergence at the boundary points \[x = 3 \pm 1\] (assuming radius 1 from the test) by substituting these values into the series and analyzing whether the resulting series converges absolutely, conditionally, or diverges, using tests like the Alternating Series Test or p-series comparison.
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Interval of Convergence
The interval of convergence is the set of all x-values for which a given power series converges. To find it, one typically uses the Ratio or Root Test to determine the radius of convergence, then checks the endpoints separately to see if the series converges there.
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08:44
Interval of Convergence
Conditional vs. Absolute Convergence
A series converges absolutely if the series of absolute values converges; otherwise, it may converge conditionally if the original series converges but not absolutely. Conditional convergence often occurs when terms alternate in sign or decrease slowly.
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Percorso guidato
07:51Choosing a Convergence Test
Behavior of the General Term and Limit Comparison
Analyzing the general term, especially expressions like (√(n+1) − √n), helps understand the series' behavior. Simplifying such terms and comparing them to known convergent or divergent series using limit comparison tests aids in determining convergence properties.
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Percorso guidato
07:45Limit Comparison Test
Pratica correlata
Domanda del libro di testo
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (c) conditionally?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]
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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
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
Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
e. Diverges for x = 8
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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) ]
29
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