Textbook Question
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 ]
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 10.7.36c
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 ]
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
07:45Limit Comparison Test
Related Practice
Textbook Question
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) ]
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)
1
views
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.)
1
views
Textbook Question
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
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) ]