Explanation Given : The sequence b n = n ! n n Calculation : Step 1: b n = n ! n n can be rewritten as b n = ( n ! ) 1 n n Step 2: Taking natural logarithms on both sides, we get ln b n = ln ( n ! ) 1 n − ln n = 1 n ln ( n ! ) − ln n = ln ( n ! ) − n ln n n = 1 n [ ln ( n ! ) − ln n n ] = 1 n ln n ! n n = 1 n ln ( 1 n ... To determine (b) The mathematical proof of the statement stating that ln ( b n ) converges to ∫ 0 1 ln x d x if b n = n ! n n and that b n → e − 1

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Videos

Question

Chapter 11.1, Problem 90E

To determine

(a)

The mathematical proof of the relation $\ln (b_{n}) = \frac{1}{n} \sum_{k = 1}^{n} \ln \frac{k}{n}$ if $b_{n} = \frac{\sqrt[n]{n!}}{n}$

To determine

(b)

The mathematical proof of the statement stating that $\ln (b_{n})$ converges to $\int_{0}^{1} \ln x d x$ if $b_{n} = \frac{\sqrt[n]{n!}}{n}$ and that $b_{n} \to e^{- 1}$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.1, Problem 89E

Chapter 11.1, Problem 91E

Students have asked these similar questions

Find the tangent line approximation 7 to the graph of f at the given point. T(x) = f(x) = csc(x), (8, csc(8)) Complete the table. (Round your answers to four decimal places.) x f(x) T(x) 7.9 7.99 8 8.01 8.1

Can you solve it numerical method

Use the information to find and compare Ay and dy. (Round your answers to four decimal places.) Function x-Value Differential of x Ду = dy = y = x² + 2 x = -4 Ax = dx = 0.01

Chapter 11 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

(a) The mathematical proof of the relation ln ( b n ) = 1 n ∑ k = 1 n ln k n if b n = n ! n n

Solution Summary: The author explains how the mathematical proof of the relation ln (b n) is mathematically proved.

Videos

(a)

The mathematical proof of the relation $\ln (b_{n}) = \frac{1}{n} \sum_{k = 1}^{n} \ln \frac{k}{n}$ if $b_{n} = \frac{\sqrt[n]{n!}}{n}$

(b)

The mathematical proof of the statement stating that $\ln (b_{n})$ converges to $\int_{0}^{1} \ln x d x$ if $b_{n} = \frac{\sqrt[n]{n!}}{n}$ and that $b_{n} \to e^{- 1}$

Want to see the full answer?

Chapter 11 Solutions

(a) The mathematical proof of the relation ln ( b n ) = 1 n ∑ k = 1 n ln k n if b n = n ! n n

Solution Summary: The author explains how the mathematical proof of the relation ln (b n) is mathematically proved.

Videos

(a) The mathematical proof of the relation ln(bn)=1n∑k=1nlnkn if bn=n!nn

(b) The mathematical proof of the statement stating that ln(bn) converges to ∫01lnxdx if bn=n!nn and that bn→e−1

Want to see the full answer?

Chapter 11 Solutions

(a)

The mathematical proof of the relation $\ln (b_{n}) = \frac{1}{n} \sum_{k = 1}^{n} \ln \frac{k}{n}$ if $b_{n} = \frac{\sqrt[n]{n!}}{n}$

(b)

The mathematical proof of the statement stating that $\ln (b_{n})$ converges to $\int_{0}^{1} \ln x d x$ if $b_{n} = \frac{\sqrt[n]{n!}}{n}$ and that $b_{n} \to e^{- 1}$