Given the power series expansion

 $\displaystyle {\ln{{\left({1}+{x}\right)}}}={\sum_{{{n}={1}}}^{{\infty}}}{\left(-{1}\right)}^{{{n}-{1}}}\frac{{x}^{{n}}}{{n}}$ 

use Taylor's Theorem with Remainder to determine how many terms $\displaystyle {N}$  of the sum evaluated at $\displaystyle {x}=-\frac{{1}}{{2}}$  are needed to approximate $\displaystyle {\ln{{\left({2}\right)}}}$  accurate to within $\displaystyle \frac{{1}}{{1000}}$.

$\displaystyle {N}=$