Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Let r > 0 be a positive real number. This problem will give a simple approach to limn→∞ rn.

Let r > 0 be a positive real number. This problem will give a simple approach to limn→∞ rn.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 91E
See similar textbooks
icon
Related questions
Question

Let r > 0 be a positive real number. This problem will give a simple approach to limn→∞ rn.
i. Explain briefly why an = rn can be written recursively as a0 = 1, an = r ⋅ an-1 for n > 0.
ii. Using Problem A and (i), solve for the possible limits of an if the sequence converges.
iii. Show that an is monotone. (It may be increasing or decreasing, depending on r.)
iv. Combine (ii) and (iii) to show limn→∞ rn converges to 0 for r < 1, converges to 1 for r = 1, and diverges to ∞ for r > 1.

Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS