Chapter 8: Q. 15 (page 669)

What is $x_{0}$ if the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ converges conditionally at both $x = - 4$ and $x = 8$ .

Short Answer

Expert verified

Ans: $x_{0} = 2$

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

role="math" localid="1649361867412" $- 1 + x_{0} < x < 1 + x_{0}$

Step 4. Thus,

Now, to find the value of $x_{0}$ , such that the series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ convergence conditionally at both role="math" localid="1649361584817" $x = - 4$ and role="math" localid="1649361594209" $x = 8$ , the interval of convergence must satisfy

role="math" localid="1649361570464" $- 1 + x_{0} = - 4 and 1 + x_{0} = 8$

Hence, solving for $x_{0}$

$x_{0} = 2$

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What is $x_{0}$ if the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ converges conditionally at both $x = - 4$ and $x = 8$ .

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 the power series ∑k=0∞ akx−x0kconverges conditionally at both x=-4and x=8.

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,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Pure Maths

Mechanics Maths

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

What is $x_{0}$ if the power series $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ converges conditionally at both $x = - 4$ and $x = 8$ .