Answer
(a) $s_{n}\leq 1+ln(n)$
(b) $s_{m}\leq 1+ln(10^{6})=1+6 ln 10\approx 14.816\lt 15$
where $m =10^{6}$ = 1 Million
and
$s_{b}\leq 1+ln(10^{9})=1+9 ln 10\approx 21.723\lt 22$
where $b =10^{9}$ = 1 Billion
Work Step by Step
(a) $\frac{1}{2}+\frac{1}{3}+....+\frac{1}{k}+...+\frac{1}{n}\lt \int_{1}^{n}\frac{dx}{x}$
$\frac{1}{2}+\frac{1}{3}+....+\frac{1}{k}+...+\frac{1}{n}\lt ln(n)$
Add $1$ to both sides.
$1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{k}+...+\frac{1}{n}\leq 1+ln(n)$
LHS is $s_{n}$
Thus, $s_{n}\leq 1+ln(n)$
(b) The equality occurs only when $n=1$
Therefore, we have $s_{m}\leq 1+ln(10^{6})=1+6 ln 10\approx 14.816\lt 15$
where $m =10^{6}$ = 1 Million
and
$s_{b}\leq 1+ln(10^{9})=1+9 ln 10\approx 21.723\lt 22$
where $b =10^{9}$ = 1 Billion
