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 |

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Chapter 11.6, Problem 1E

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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 $x = \pm R$ .

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Chapter 11.6, Problem 4PQ

Chapter 11.6, Problem 2E

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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

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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.

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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 $x = \pm R$ .