Q. 15 What is x0聽if the power series ... [FREE SOLUTION]

91影视

Chapter 8: Q. 15 (page 669)

What is $x_{0}$ if the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k}$ 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,

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

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

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

Step 4. Thus,

Now, to find the value of $x_{0}$ , such that the series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k}$ 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" $鈭� 1 + x_{0} = 鈭� 4 and 1 + x_{0} = 8$

Hence, solving for $x_{0}$

$x_{0} = 2$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

What is $x_{0}$ if the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k}$ 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,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

91影视

What is x0if the power series 鈭�k=0鈭�鈥�akx鈭�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

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

What is $x_{0}$ if the power series ${鈭�}_{k = 0}^{鈭�} 鈥� a_{k} {(x 鈭� x_{0})}^{k}$ converges conditionally at both $x = - 4$ and $x = 8$ .