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How do you find the interval of convergence of a Taylor

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 4E Chapter 9.3

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 4E

Problem 4E

How do you find the interval of convergence of a Taylor series?

Step-by-Step Solution:

Solution 4E

Step 1:

General  expression of Taylor series of  a function f(x) centered at ‘a’ is ;

              = f(a) +(x-a) +++.............

Consider , = .

  contains factorial notation . So , in this cases we can use  ‘ratio test ‘ to find the  interval convergence of a Taylor series.

By , the ratio test ;

If  P = | | < 1 ,  then the series converges.

And if  P > 1 , then the series diverges .

If P = 1, then the test is inconclusive.

Consider, P = |x|

The ratio test tells us now that the series  will converges as long as |x| < 1 . It also tells us that the series will diverges for |x| > 1.

The biggest interval ( it is always an interval ) where a Taylor series is convergent is called interval of convergence of the Taylor series . The interval of convergence is always centered at the center of  the Taylor series .

             

           

Step 2 of 1

Chapter 9.3, Problem 4E is Solved
Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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How do you find the interval of convergence of a Taylor