Q. 13 What is x0聽if the interval of c... [FREE SOLUTION]

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Chapter 8: Q. 13 (page 669)

What is $x_{0}$ if the interval of convergence for the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k} is (2, 10] ?$

Short Answer

Expert verified

Ans: $x_{0} = 6$

Step by step solution

Step 1. Given information.

given,

${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k}$

Step 2. To find the interval of convergence for the power series, let us first assume bk=akx−x0k, so bk+1=ak+1x−x0k+1

Therefore,

$\begin{aligned} lim_{k 鈫� 鈭�} 鈥� |\frac{b_{k + 1}}{b_{k}}| = lim_{k 鈫� 鈭�} 鈥� |\frac{a_{k + 1} {(x 鈭� x_{0})}^{k + 1}}{a_{k} {(x 鈭� x_{0})}^{k}}| \\ = lim_{k 鈫� 鈭�} 鈥� \frac{a_{k + 1}}{a_{k}} |x 鈭� x_{0}| \\ = \frac{a_{k + 1}}{a_{k}} lim_{k 鈫� 鈭�} 鈥� |x 鈭� x_{0}| \end{aligned}$

Step 3. Now,

Here the limit is $|x 鈭� x_{0}|$ . So, by the ratio test of absolute convergence, we know that the series will converge absolutely, when $|x 鈭� x_{0}| < 1$ , that is $- 1 < x - x_{0} < 1$

Implies that

$鈭� 1 + x_{0} < x < 1 + x_{0}$

Step 4. Thus,

Now, to find the value of $x_{0}$ , such that the interval of convergence of the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k}$ is $(2, 10]$ , the interval of convergence must satisfy $鈭� 1 + x_{0} = 2 and 1 + x_{0} = 10$

Hence, solving for $x_{0}$

$x_{0} = 6$

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91影视

What is $x_{0}$ if the interval of convergence for the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k} is (2, 10] ?$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. To find the interval of convergence for the power series, let us first assume bk=akx−x0k, so bk+1=ak+1x−x0k+1

Step 3. Now,

Step 4. Thus,

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91影视

What is x0if the interval of convergence for the power series 鈭�k=0鈭�鈥�akx鈭�x0kis(2,10]?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. To find the interval of convergence for the power series, let us first assume bk=akx−x0k, so bk+1=ak+1x−x0k+1

Step 3. Now,

Step 4. Thus,

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

What is $x_{0}$ if the interval of convergence for the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k} is (2, 10] ?$