Chapter 8: Q. 14 (page 669)

What is $x_{0}$ if $(p, q)$ is the interval of convergence for the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ ?

Short Answer

Expert verified

Ans: $x_{0} = \frac{p + q}{2}$

Step by step solution

Step 1. Given information.

given,

$\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$

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,

$\begin{aligned} lim_{k \to \infty} |\frac{b_{k + 1}}{b_{k}}| = lim_{k \to \infty} |\frac{a_{k + 1} {(x - x_{0})}^{k + 1}}{a_{k} {(x - x_{0})}^{k}}| \\ = lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} |x - x_{0}| \\ = \frac{a_{k + 1}}{a_{k}} lim_{k \to \infty} |x - x_{0}| \end{aligned}$

Step 3. Now,

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

Implies that

$- 1 + x_{0} < x < 1 + x_{0}$

Step 4. Thus,

Now, to find the value of $x_{0}$ , such that the interval of convergence of the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ is $(p, q)$ , the interval of convergence must satisfy

$- 1 + x_{0} = p and 1 + x_{0} = q$

Hence, solving for $x_{0}$

$x_{0} = \frac{p + q}{2}$

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What is $x_{0}$ if $(p, q)$ is the interval of convergence for the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ ?

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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What is x0if (p,q)is the interval of convergence for the power series ∑k=0∞ akx−x0k?

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

Statistics

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Theoretical and Mathematical Physics

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Logic and Functions

Study anywhere. Anytime. Across all devices.

What is $x_{0}$ if $(p, q)$ is the interval of convergence for the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ ?