Chapter 9.4, Problem 40ES

Exercise 36 will show how a partial sum can be used to obtain upper and lower bounds on the sum of a series when the hypotheses of the integral test are satisfied. This result will be needed in Exercises 37-40.

We showed in Section 9.3 that the harmonic series ${\displaystyle {\sum }_{k=1}^{\infty }1/k}$ diverges. Our objective in this problem is to demonstrate that although the partial sums of this series approach $+\infty ,$ they increase extremely slowly.

(a) Use inequality (2) to show that for $n\ge 2$

$\ln \left(n+1\right)<s_n<1+\ln n$

(b) Use the inequalities in part (a) to find upper and lower bounds on the sum of the first million terms in the series.

(c) Show that the sum of the first billion terms in the series is less than 22.

(d) Find a value of $n$ so that the sum of the first n terms is greater than 100.