To calculate: The series converges absolutely, converges conditionally, or diverges.
Answer to Problem 25E
The series converges conditionally, and the truncation error is less than
Explanation of Solution
Given information:
Calculation:
Let examine whether the series converges absolutely. If it does not, let use other tests in order to determine conditional convergence. The bound for the truncation error can be determined by using the Alternating Series Bound Theorem.
First let examine the series
In order to determine the convergence or divergence of the series let use the Integral Test.
Based on Integral Test,
Let evaluate the improper integral from the previous step. Let
because
Based on the previous step,
Let check whether the series converges conditionally by using the alternating series test.
Since
The expression
Finally,
Since
converges conditionally.
The truncation error can be determined by using the Alternating Series Bound Theorem. Based on this, the error after 99 terms is less than
Therefore, the series converges conditionally, and the truncation error is less than
Chapter 9 Solutions
CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS
- 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