Chapter 5, Problem 118CRE

To determine

(a)

To show:

That $\ln \left(1+x\right)={\displaystyle \underset{0}{\overset{x}{\int }}\frac{dt}{1+t}}$ for $x>0$.

To determine

(b)

To verify:

The inequality $1-t\le \frac{1}{1+t}\le 1$ for $t>0$.

To determine

(c)

To prove:

The inequality $x-\frac{x^2}{2}\le \ln \left(1+x\right)\le x$ by use of part (b).

To determine

(d)

To verify:

The given inequality holds for $x=0.5,0.1,0.01$ .