To state: Why the same conclusions about convergence can be drawn if the picture of an arbitrary N is relabeled.
Answer to Problem 53E
If the integral diverges, it must go to infinity, and the first inequality forces the partial sums of the series to go to infinity as well, so the series is divergent.
Explanation of Solution
Given information:
The given statement says to explain the reason about why the same conclusions about convergence can be drawn if the picture of an arbitrary N is relabeled.
The graphs are drawn for an arbitrary N as follows:
Comparing areas in the figures, it must be for all
If the integral diverges, it must go to infinity, and the first inequality forces the partial sums of the series to go to infinity as well, so the series is divergent.
If the integral converges, then the second inequality puts an upper bound on the partial sums to the series, and since they are a no decreasing sequence, they must converge to a finite sum for the series.
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