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Let a, = Hn – Inn, where H, is the nth harmonic number: 89. 1. 1 H, = 1++ п dx (a) Show that a, 2 0 for n > 1. Hint: Show that H, 2 (b) Show that {a,} is decreasing by interpreting a, – an+1 as an area. (c) Prove that lim a, exists. This limit, denoted y, is known as Euler's Constant. It appears in many areas of mathematics, including analysis and number theory, and has been calculated to more than 100 million decimal places, but it is still not known whether y is an irrational number. The first 10 digits are y 0.5772156649. х

Let a, = Hn – Inn, where H, is the nth harmonic number: 89. 1. 1 H, = 1++ п dx (a) Show that a, 2 0 for n > 1. Hint: Show that H, 2 (b) Show that {a,} is decreasing by interpreting a, – an+1 as an area. (c) Prove that lim a, exists. This limit, denoted y, is known as Euler's Constant. It appears in many areas of mathematics, including analysis and number theory, and has been calculated to more than 100 million decimal places, but it is still not known whether y is an irrational number. The first 10 digits are y 0.5772156649. х

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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Let a, = Hn – Inn, where H, is the nth harmonic number: 89. 1. 1 H, = 1++ п dx (a) Show that a, 2 0 for n > 1. Hint: Show that H, 2 (b) Show that {a,} is decreasing by interpreting a, – an+1 as an area. (c) Prove that lim a, exists. This limit, denoted y, is known as Euler's Constant. It appears in many areas of mathematics, including analysis and number theory, and has been calculated to more than 100 million decimal places, but it is still not known whether y is an irrational number. The first 10 digits are y 0.5772156649. х
Transcribed Image Text:Let a, = Hn – Inn, where H, is the nth harmonic number: 89. 1. 1 H, = 1++ п dx (a) Show that a, 2 0 for n > 1. Hint: Show that H, 2 (b) Show that {a,} is decreasing by interpreting a, – an+1 as an area. (c) Prove that lim a, exists. This limit, denoted y, is known as Euler's Constant. It appears in many areas of mathematics, including analysis and number theory, and has been calculated to more than 100 million decimal places, but it is still not known whether y is an irrational number. The first 10 digits are y 0.5772156649. х
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