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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

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter8: Areas Of Polygons And Circles
Section8.4: Cicumference And Area Of A Cicle
Problem 21E: Let N be any point on side BC of the right triangle ABC. Find the upper and lower limits for the...
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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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