Problem $2.$

Let $f$:$(0,)R$ be the logarithm function, i$.$e$.,f(x)=lnx.$ As we have not defined this as an infinite series yet, we shall assume $($without proof$)$ that $f$ satisfies the following properties:

$($a$)f(1)=0$

$($b$)f$ is differentiable and the derivative is the function $\frac{1}{x}.$

Using Taylor's Theorem prove that

$x>0$kinNx$-\frac{1}{2}{x}^2+$cdots$-\frac{1}{2k}{x}^{2k}$

for any $x>0$ and any kinN.