Chapter 8: Q. 4 (page 692)

If the series $\sum_{k = 0}^{\infty} b_{k} {(x - 3)}^{k}$ converges to the function $g (x)$ on the interval (−2, 8), provide a formula for $b_{k}$ in terms of the function g.

Short Answer

Expert verified

The formula for $b_{k}$ is $\frac{f^{(k)} (3)}{k!}$ .

Step by step solution

Step 1. Given Information.

The series $\sum_{k = 0}^{\infty} b_{k} {(x - 3)}^{k}$ is converges to $g (x)$ on $(- 2, 8)$ .

Step 2. Explanation.

The series is converges to the function so, Taylor polynomial for $g (x)$ is in the form role="math" localid="1649488146257" $\sum_{k = 0}^{\infty} \frac{f^{(k)} x_{0}}{k!} (x - x_{0})$ .

For series $\sum_{k = 0}^{\infty} b_{k} {(x - 3)}^{k}$ , So, $b_{k} = \frac{f^{k} (3)}{k!}$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

If the series $\sum_{k = 0}^{\infty} b_{k} {(x - 3)}^{k}$ converges to the function $g (x)$ on the interval (−2, 8), provide a formula for $b_{k}$ in terms of the function g.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Decision Maths

Mechanics Maths

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

If the series ∑k=0∞bkx-3kconverges to the function g(x) on the interval (−2, 8), provide a formula for bk in terms of the function g.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Decision Maths

Mechanics Maths

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

If the series $\sum_{k = 0}^{\infty} b_{k} {(x - 3)}^{k}$ converges to the function $g (x)$ on the interval (−2, 8), provide a formula for $b_{k}$ in terms of the function g.