a. Sketch the function f(x) = 1/x on the interval [1, n + 1], where n is

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a. Sketch the function f(x) = 1/x on the interval [1, n + 1], where n is a positive integer. Use this graph to verify that

1 In (n + 1) < 1 + 1 1 < 1 + In n.

b. Let Sn be the sum of the first n terms of the harmonic series, so part (a) says ln (n + 1) < Sn < 1 + ln n. Define the new sequence {En} by 

En = Sn - ln (n + 1), for n = 1, 2, 3, . . . Show that En > 0, for n = 1, 2, 3,  . . .

c. Using a figure similar to that used in part (a), show that

> In (n + 2) – In (n + 1). n + 1

d. Use parts (a) and (c) to show that {En} is an increasing sequence (En + 1 > En).

e. Use part (a) to show that {En} is bounded above by 1.

f. Conclude from parts (d) and (e) that {En} has a limit less than or equal to 1. This limit is known as Euler’s constant and is denoted γ (the Greek lowercase gamma).

g. By computing terms of {En}, estimate the value of γ and compare it to the value γ ≈ 0.5772. (It has been conjectured that γ is irrational.)

h. The preceding arguments show that the sum of the first n terms of the harmonic series satisfy Sn ≈ 0.5772 + ln (n + 1). How many terms must be summed for the sum to exceed 10?

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Calculus Early Transcendentals

ISBN: 978-0321947345

2nd edition

Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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