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.

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

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.

Videos

Question

Chapter 11.1, Problem 93E

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.1, Problem 92E

Chapter 11.2, Problem 1PQ

Students have asked these similar questions

Decide whether each limit exists. If a limit exists, estimate its value. 11. (a) lim f(x) x-3 f(x) ↑ 4 3- 2+ (b) lim f(x) x―0 -2 0 X 1234

Determine whether the lines L₁ (t) = (-2,3, −1)t + (0,2,-3) and L2 p(s) = (2, −3, 1)s + (-10, 17, -8) intersect. If they do, find the point of intersection.

Convert the line given by the parametric equations y(t) Enter the symmetric equations in alphabetic order. (x(t) = -4+6t = 3-t (z(t) = 5-7t to symmetric equations.

Chapter 11 Solutions

CALCULUS (CLOTH)

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 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

Videos

(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

Videos

(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.