Q. 69 聽Let b聽be a nonzero constant. ... [FREE SOLUTION]

Chapter 8: Q. 69 (page 671)

Let $b$ be a nonzero constant. Prove that the radius of convergence of the power series ${鈭�}_{k = 0}^{鈭�} 鈥� b^{k} x^{k} is \frac{1}{| b |}$ .

Short Answer

Expert verified

Ans: It is proved that the radius of convergence of the power series ${鈭�}_{k = 0}^{鈭�} 鈥� b^{k} x^{k} is \frac{1}{| b |}$

Step by step solution

Step 1. Given information.

given,

${鈭�}_{k = 0}^{鈭�} 鈥� b^{k} x^{k} is \frac{1}{| b |}$

Step 2. Consider the power series ∑k=0∞ bkxk , where b be a non-zero constant.

Also, let us consider $b_{k} = a_{k} x^{k}, so b_{k + 1} = a_{k + 1} x^{k + 1}$

Apply the ratio test for absolute convergence in the power series ${鈭�}_{k = 0}^{鈭�} 鈥� b^{k} x^{k}$ , that is

$\begin{aligned} lim_{k 鈫� 鈭�} 鈥� |\frac{b_{k + 1}}{b_{k}}| = lim_{k 鈫� 鈭�} 鈥� |\frac{b^{k + 1}}{b^{k}} x| \\ = lim_{k 鈫� 鈭�} 鈥� | b x | \end{aligned}$

So according to the ratio test for absolute convergence, the series will converge only when $| b x | < 1$

Implies that $| x | < \frac{1}{| b |}$

Step 3. Thus,

The radius of convergence of the power series ${鈭�}_{k = 0}^{鈭�} 鈥� b^{k} x^{k} is \frac{1}{| b |}$

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Let $b$ be a nonzero constant. Prove that the radius of convergence of the power series ${鈭�}_{k = 0}^{鈭�} 鈥� b^{k} x^{k} is \frac{1}{| b |}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the power series ∑k=0∞ bkxk , where b be a non-zero constant.

Step 3. Thus,

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91影视

Let bbe a nonzero constant. Prove that the radius of convergence of the power series 鈭�k=0鈭�鈥�bkxkis1|b|.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the power series ∑k=0∞ bkxk , where b be a non-zero constant.

Step 3. Thus,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Logic and Functions

Calculus

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

Let $b$ be a nonzero constant. Prove that the radius of convergence of the power series ${鈭�}_{k = 0}^{鈭�} 鈥� b^{k} x^{k} is \frac{1}{| b |}$ .