Explanation Given : The sequence H n = 1 + 1 2 + 1 3 + ... + 1 n and a n = H n − ln n Calculation : Step 1: Since the function y = 1 x is decreasing , the left endpoint approximation to the integral ∫ 1 n + 1 d x x is greater than this integral. Hence, 1.1 + 1 2 .1 + 1 3 .1 + ... + 1 n .1 ≥ ∫ 1 n + 1 d x x or H n ≥ ∫ 1 n + 1 d x x To determine (b) The mathematical proof of the statement stating that { a n } is decreasing by interpreting a n − a n + 1 as an area. To determine (c) The mathematical proof of the statement stating that lim n → ∞ a n exists.

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 11.1, Problem 89E

To determine

(a)

The mathematical proof of the statement stating that $a_{n} \geq 0$ given that $a_{n} = H_{n} - \ln n$ where $H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$

To determine

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is decreasing by interpreting $a_{n} - a_{n + 1}$ as an area.

To determine

(c)

The mathematical proof of the statement stating that $\lim_{n \to \infty} a_{n}$ exists.

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Chapter 11.1, Problem 88E

Chapter 11.2, Problem 1PQ

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(a) The mathematical proof of the statement stating that a n ≥ 0 given that a n = H n − ln n where H n = 1 + 1 2 + 1 3 + ... + 1 n

Solution Summary: The author analyzes the mathematical proof of the statement stating that a n 0 is mathematically proved. Since the function y = 1 x is decreasing, the left endpoint approximation to the integral

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(a)

The mathematical proof of the statement stating that $a_{n} \geq 0$ given that $a_{n} = H_{n} - \ln n$ where $H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is decreasing by interpreting $a_{n} - a_{n + 1}$ as an area.

(c)

The mathematical proof of the statement stating that $\lim_{n \to \infty} a_{n}$ exists.

Want to see the full answer?

Chapter 11 Solutions

(a) The mathematical proof of the statement stating that a n ≥ 0 given that a n = H n − ln n where H n = 1 + 1 2 + 1 3 + ... + 1 n

Solution Summary: The author analyzes the mathematical proof of the statement stating that a n 0 is mathematically proved. Since the function y = 1 x is decreasing, the left endpoint approximation to the integral

Concept explainers

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(a) The mathematical proof of the statement stating that an≥0 given that an=Hn−lnn where Hn=1+12+13+...+1n

(b) The mathematical proof of the statement stating that {an} is decreasing by interpreting an−an+1 as an area.

(c) The mathematical proof of the statement stating that limn→∞an exists.

Want to see the full answer?

Chapter 11 Solutions

(a)

The mathematical proof of the statement stating that $a_{n} \geq 0$ given that $a_{n} = H_{n} - \ln n$ where $H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$

(b)

The mathematical proof of the statement stating that ${a_{n}}$ is decreasing by interpreting $a_{n} - a_{n + 1}$ as an area.

(c)

The mathematical proof of the statement stating that $\lim_{n \to \infty} a_{n}$ exists.