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prove that an is bounded above.

prove that an is bounded above.

Advanced Engineering Mathematics
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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Now prove that an is bounded above. This, too, is subtle. • The Binomial Theorem tells us that п(п - 1) -x² + 2! п(n - 1)(п — 2) (1+ x)" = 1+ nx + +... 3! п(n - 1) -- (п - (п — 1)) (п — 1)! n! .. Deduce that n (1+)" 1 1 <1+1+ 2! (n – 1)! 3! n! • Show that k! for any positive integer k. 2k • Deduce that n (1+ :) <1+5 2k k=0 and that the partial sum is < 2. Thus an < 3 for all n. The Monotone Convergence Theorem tells us that the sequence converges to a limit < 3. In fact, that limit is e.
Transcribed Image Text:Now prove that an is bounded above. This, too, is subtle. • The Binomial Theorem tells us that п(п - 1) -x² + 2! п(n - 1)(п — 2) (1+ x)" = 1+ nx + +... 3! п(n - 1) -- (п - (п — 1)) (п — 1)! n! .. Deduce that n (1+)" 1 1 <1+1+ 2! (n – 1)! 3! n! • Show that k! for any positive integer k. 2k • Deduce that n (1+ :) <1+5 2k k=0 and that the partial sum is < 2. Thus an < 3 for all n. The Monotone Convergence Theorem tells us that the sequence converges to a limit < 3. In fact, that limit is e.
Expert Solution
Step 1

In this question, we prove the sequence 

an=(1+1n)n

is convergent by given porcedure.

i.e. 

monotonic convergence theorem

steps

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