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.
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.
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
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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.
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