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a. Calculate the first 5 terms of the sequence an = (1+1)", expressing your answers as fractions and as decimals to 3dp. b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as follows.

a. Calculate the first 5 terms of the sequence an = (1+1)", expressing your answers as fractions and as decimals to 3dp. b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as follows.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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a. Calculate the first 5 terms of the sequence an as fractions and as decimals to 3dp. b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as follows. Note that an = Deduce that n+1 = (₁ + ² ) *** / (₁+²). . Hence, use algebra to show that an+1 an • Deduce that Bernoulli's inequality states that for any x ≥-1, (1+x)" ≥ 1+nx. Use this to deduce that (1+x)"=1+n+ 1 Show that < k! and simplify the last expression. c. Now prove that an is bounded above. This, too, is subtle. The Binomial Theorem tells us that n(n-1) 2! 1 (₁+²)" 1+ 1 2k n - (₁ + ²) "₁ n+1 1 = - (1 - (n +² 1²) "* ¹ (1+²) an+1 1 dan 1 ≥ (1 - 2 + ₁) (₁ + ² ) > n+1 +... , expressing your answers + n(n-1)(n-2) 3! -23 n(n − 1) ... ((n − (n − 1))n- (n − 1)! 1 1 < 1+1+ + + + 2! 3! for any integer k ≥ 4. (₁+1) "< + £ 12/10 2k k=0 and that the partial sum is <2. Thus an < 3 for all n. +... + 1 1 + (n 1)! n! n! 21 22 The Monotone Convergence Theorem tells us that the sequence converges to a limit <3. In fact, that limit is e.
Transcribed Image Text:a. Calculate the first 5 terms of the sequence an as fractions and as decimals to 3dp. b. Prove that the sequence is increasing. Actually this is a bit hard, but proceed as follows. Note that an = Deduce that n+1 = (₁ + ² ) *** / (₁+²). . Hence, use algebra to show that an+1 an • Deduce that Bernoulli's inequality states that for any x ≥-1, (1+x)" ≥ 1+nx. Use this to deduce that (1+x)"=1+n+ 1 Show that < k! and simplify the last expression. c. Now prove that an is bounded above. This, too, is subtle. The Binomial Theorem tells us that n(n-1) 2! 1 (₁+²)" 1+ 1 2k n - (₁ + ²) "₁ n+1 1 = - (1 - (n +² 1²) "* ¹ (1+²) an+1 1 dan 1 ≥ (1 - 2 + ₁) (₁ + ² ) > n+1 +... , expressing your answers + n(n-1)(n-2) 3! -23 n(n − 1) ... ((n − (n − 1))n- (n − 1)! 1 1 < 1+1+ + + + 2! 3! for any integer k ≥ 4. (₁+1) "< + £ 12/10 2k k=0 and that the partial sum is <2. Thus an < 3 for all n. +... + 1 1 + (n 1)! n! n! 21 22 The Monotone Convergence Theorem tells us that the sequence converges to a limit <3. In fact, that limit is e.
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