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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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