(a) The mathematical proof of the relation a n ≤ b n for all n if a n + 1 = a n b n , b n + 1 = a n + b n 2 given that the positive numbers a 1 < b 1

Question

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Answer 1

Question

Chapter 10.1, Problem 91E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

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Class, the class silues, and the class notes, whether the series does alternate and the absolute values of the terms decrease), and if the test does apply, determine whether the series converges or diverges. For the ith series, if the test does not apply the let Mi = 2, while if the test determines divergence then M¿ = 4, and if it determines convergence then M¿ = 8. 1: 2: 3 : 4: 5 : ∞ n=1 ∞ (−1)n+1. Σ(-1) +1 n=1 ∞ п 3m² +2 Σ(-1)+1 sin(2n). n=1 ∞ 2n² + 2n +3 4n2 +6 1 e-n + n² 3n23n+1 9n² +3 In(n + 1) 2n+1 Σ(-1) +1 n=1 ∞ Σ(-1)". n=1 Then the value of cos(M₁) + cos(2M2) + cos(3M3) + sin(2M4) + sin(M5) is 1.715 0.902 0.930 -1.647 -0.057 ● 2.013 1.141 4.274

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Answer 2

Question

Chapter 10.1, Problem 91E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

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Answer 3

Question

Chapter 10.1, Problem 91E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

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Answer 4

Question

Chapter 10.1, Problem 91E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 5

Chapter 10.1, Problem 91E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Answer 6

Chapter 10.1, Problem 91E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

Answer 7

Chapter 10.1, Problem 91E

To determine

(a)

The mathematical proof of the relation $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$

Answer 8

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

Answer 9

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is increasing and ${b_{n}}$ is decreasing $a_{n} \leq b_{n}$ for all n if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$ .

Answer 10

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

Answer 11

To determine

(c)

The mathematical proof of the relation $b_{n + 1} - a_{n + 1} \leq \frac{b_{n} - a_{n}}{2}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given positive numbers $a_{1} < b_{1}$

Answer 12

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

Answer 13

To determine

(d)

The mathematical proof of the statement stating that both ${a_{n}}$ and ${b_{n}}$ converge and have the same limit which is denoted by $A G M (a_{1}, b_{1})$ and which is called the arithmetic-geometric mean of $a_{1}$ and $b_{1}$ if $a_{n + 1} = \sqrt{a_{n} b_{n}}, b_{n + 1} = \frac{a_{n} + b_{n}}{2}$ given that the positive numbers $a_{1} < b_{1}$ .

Answer 14

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Answer 15

To determine

(e)

The estimated value of AGM (1, $\sqrt{2})$ to three decimal places.

Answer 16

Expert Solution & Answer

Answer 17

Expert Solution & Answer

Answer 18

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Check out a sample textbook solution

See solution

Answer 19

Ch. 10.1 - Prob. 1PQ Ch. 10.1 - Prob. 2PQ Ch. 10.1 - Prob. 3PQ Ch. 10.1 - Prob. 4PQ Ch. 10.1 - Prob. 5PQ Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E

(a) The mathematical proof of the relation a n ≤ b n for all n if a n + 1 = a n b n , b n + 1 = a n + b n 2 given that the positive numbers a 1 &lt; b 1