   Chapter 11.3, Problem 43E

Chapter
Section
Textbook Problem

# (a) Use (4) to show that if sn 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.

(a)

To determine

To show: The nth partial sum of the harmonic series sn1+lnn .

Explanation

Result used: Remainder Estimate for the Integral Test

If the function f(x) is continuous, positive, and decreasing for xn and an is convergent and Rn=ssn , then n+1f(x)dxRnnf(x)dx .

Proof:

The nth partial sum of the harmonic series n=11n is, sn=1+12+13+14+...+1n . (1)

Sketch the graph of the function f(x)=1x .

From Figure 1, it is observed that the total area of the rectangles is less than the area under f(x) of the curve.

Compare the areas of the shaded rectangles with the area under y=1x from 1 to n from Figure 1 with the Remainder Estimate for the Integral Test.

That is, a2+a3+...+an1nf(x)dx

12+13+14+

(b)

To determine

To show: The sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22 by using part (a).

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