To prove: The Maclaurin series for the given function converges for all real x using remainder estimation theorem .
Explanation of Solution
Given:
Function is
Proof:
As,
So, to show that the Maclaurin series for
Now,
Using Remainder theorem, it states that
Here M and r are constants,
Here, a = 0 as we are dealing with Maclaurin series
Since
Either
Which means that
So, Set M and r equal to 1.
Then,
Since
Also
Since factorials increase faster than exponentials. This can also be done using the Ratio test.
So, conditions are satisfied
Chapter 10 Solutions
Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)
Additional Math Textbook Solutions
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
University Calculus: Early Transcendentals (3rd Edition)
Precalculus Enhanced with Graphing Utilities (7th Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
Calculus and Its Applications (11th Edition)
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning