91Ó°ÊÓ

Prove that $$ 1+\frac{x}{2}-\frac{x^{2}}{8}<\sqrt{1+x}<1+\frac{x}{2} \quad \text { if } x>0 $$ In particular, show that \(1.375<\sqrt{2}<1.5\).

Apply the Cauchy Integral Remainder Theorem in the analysis of the expansion $$ \ln (1+x)=\sum_{k=1}^{\infty}(-1)^{k+1} \frac{x^{k}}{k} \quad \text { if }-1