Explanation The Radio Test for series is very useful test for determining whether or not a series converges or diverges. The Ratio Test says that: Let the following limit exists, L = lim n → ∞ | a n + 1 a n | I. If L < 1 , then the original series ∑ n = 1 ∞ a n will be absolutely convergent. II. If L > 1 or L = ∞ , then the original series ∑ n = 1 ∞ a n will be divergent. III. If L = 1 , then no conclusion can be made (the series may diverge or converge). Given: ∑ n = 0 ∞ x n 2 n Formula Used: L = lim n → ∞ | a n + 1 a n | Calculation: The given series is: ∑ n = 0 ∞ x n 2 n and we have to find the radius of convergence. So, let a n = x n 2 n . L = lim n → ∞ | a n + 1 a n | = lim n → ∞ | x n + 1 2 n + 1 x n 2 n |

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Videos

Question

Chapter 11.6, Problem 1E

To determine

To calculate:

The radius of convergence R of $\sum_{n = 0}^{\infty} \frac{x^{n}}{2^{n}}$ by using the Radio Test. Also, if it converge at the endpoints $x = \pm R$ .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.6, Problem 4PQ

Chapter 11.6, Problem 2E

Students have asked these similar questions

The number of fish swimming upstream to spawn is approximated by the function given below, where x represents the temperature of the water in degrees Celsius. Find the water temperature that produces the maximum number of fish swimming upstream. F(x) = x3 + 3x² + 360x + 5017, 5≤x≤18

A campground owner has 500 m of fencing. He wants to enclose a rectangular field bordering a river, with no fencing along the river. (See the sketch.) Let x represent the width of the field. (a) Write an expression for the length of the field as a function of x. (b) Find the area of the field (area = length x width) as a function of x. (c) Find the value of x leading to the maximum area. (d) Find the maximum area. x River

A rectangular tank with a square base, an open top, and a volume of 1372 ft³ is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area. The dimensions of the tank with minimum surface area are (Simplify your answer. Use a comma to separate answers.) ft.

Chapter 11 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The radius of convergence R of ∑ n = 0 ∞ x n 2 n by using the Radio Test. Also, if it converge at the endpoints x = ± R .

Solution Summary: The author explains that the Radio Test for series is useful for determining whether or not a series converges or diverges.

Videos

To calculate:

The radius of convergence R of $\sum_{n = 0}^{\infty} \frac{x^{n}}{2^{n}}$ by using the Radio Test. Also, if it converge at the endpoints $x = \pm R$ .

Want to see the full answer?

Chapter 11 Solutions

The radius of convergence R of ∑ n = 0 ∞ x n 2 n by using the Radio Test. Also, if it converge at the endpoints x = ± R .

Solution Summary: The author explains that the Radio Test for series is useful for determining whether or not a series converges or diverges.

Videos

To calculate: The radius of convergence R of ∑n=0∞xn2n by using the Radio Test. Also, if it converge at the endpoints x=±R.

Want to see the full answer?

Chapter 11 Solutions

To calculate:

The radius of convergence R of $\sum_{n = 0}^{\infty} \frac{x^{n}}{2^{n}}$ by using the Radio Test. Also, if it converge at the endpoints $x = \pm R$ .