Chapter 9.3, Problem 65E

Euler's Constant Let

$s_n={\displaystyle \underset{k=1}{\overset{n}{\sum }}\frac{1}{k}=1+\frac{1}{2}+\mathrm{...}+\frac{1}{n^{}}}$.

(a) Show that $\ln (n+1)\le s_n\le 1+l\text{n }n.$

(b) Show that the sequence $\left\{a_n\right\}=\{s_n-\ln n\}$ is bounded.

(c) Show that the sequence $\left\{a_n\right\}$ is decreasing.

(d) Show that the sequence $\left\{a_n\right\}$ converges to a limit y (called Euler’s constant).

(e) Approximate y using $a_{100}$