   Chapter 11.3, Problem 43E

Chapter
Section
Textbook Problem

(a) Use (4) to show that if s n is the nth partial sum of the harmonic series, then s n ≤ 1 + ln n (b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22.

To determine

(a)

To show:

If sn is the nth partial sum of the harmonic series then

sn1+lnn

Explanation

1) Concept:

Compare the areas of shaded rectangles with the area under y=f(x). The inequality depends on the fact that f is decreasing

2) Calculation:

From the figure,

The area of the first shaded rectangle is value of f at the right endpoints of [1, 2]. That is

f2=a2.

So, comparing the areas of the shaded rectangles under the curve y=f(x) from 1 to n it is observed that,

a2+ ..an1nf(x)dx

So with, fx=1x

12+13+14+

To determine

(b)

To show:

The sum of the first million terms of harmonic series is less than 15, and the sum of the first billion terms of harmonic series is less than 22

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