Explanation Given that, the function is f ( x ) = 5 2 − x . Notice that, Taylor series for the function 1 1 − x is 1 + x + x 2 + x 3 + ⋯ + x n + ⋯ and its interval of convergence is ( − 1 , 1 ) . The given function 5 2 − x can be written as ( 5 2 ) 1 − ( x 2 ) . Assume that a function f has a Taylor series such that f ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ + a n x n + ⋯ . Then replace x with g ( x ) = c x k for some constant c and positive integer k , which on solving gives the Taylor series for f [ g ( x ) ] as, a 0 + a 1 g ( x ) + a 2 [ g ( x ) ] 2 + a 3 [ g ( x ) ] 3 + ⋯ + a n [ g ( x ) ] n + ⋯ . In this case the interval of convergence of new series can vary. For a real number c the Taylor series for c f ( x ) is c ⋅ a 0 + c ⋅ a 1 x + c ⋅ a 2 x 2 + ⋯ + c ⋅ a n x n + ⋯ for all x in the interval of convergence

Calculus with Applications Books a la Carte Edition

11th Edition

ISBN: 9780133864564

Author: Margaret L. Lial; Nathan P. Ritchey; Raymond N. Greenwell

Publisher: Pearson Education

See similar textbooks

Videos

Question

Chapter 12.5, Problem 5E

To determine

To find: The Taylor series for the function f(x)=52−x and also find its interval of convergence.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.5, Problem 4E

Chapter 12.5, Problem 6E

Students have asked these similar questions

Good Day, Kindly assist me with the following query. Any assistance would be appreciated.

Can u give rough map of any room u can choose cm on top

3. We'd like to know the first time when the population reaches 7000 people. First, graph the function from part (a) on your calculator or Desmos. In the same window, graph the line y = 7000. Notice that you will need to adjust your window so that you can see values as big as 7000! Investigate the intersection of the two graphs. (This video shows you how to find the intersection on your calculator, or in Desmos just hover the cursor over the point.) At what value t> 0 does the line intersect with your exponential function? Round your answer to two decimal places. (You don't need to show work for this part.) (2 points)

Chapter 12 Solutions

Calculus with Applications Books a la Carte Edition

Ch. 12.1 - Find the first four terms of the sequence having...Ch. 12.1 - Prob. 2YT Ch. 12.1 - Prob. 3YT Ch. 12.1 - Prob. 4YT Ch. 12.1 - Prob. 5YT Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - List the first n terms of the geometric sequence...Ch. 12.1 - Prob. 5E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The Taylor series for the function f ( x ) = 5 2 − x and also find its interval of convergence.

Videos

To find: The Taylor series for the function f(x)=52−x and also find its interval of convergence.

Want to see the full answer?

Chapter 12 Solutions