\(\displaystyle \sqrt6 \ = \ 2.44948974278.... \ (TI-89).\)
\(\displaystyle \sqrt6 \ = \ 2\sqrt(1+\frac{1}{2}) \ = \ 2(1+\frac{1}{2})^{1/2}.\)
\(\displaystyle Hence, \ with \ only \ four \ iterations, \ we \ get:\)
\(\displaystyle 2[1+.25+\frac{[(.5)(-.5)(.25)]}{2}+\frac{[(.5)(-.5)(-1.5)(.125)]}{6}] \ = \ 2.453625\)
\(\displaystyle 2.453625-2.44948974278 \ = \ .00363525722, \ need \ about \ two \ more \ iterations.\)
\(\displaystyle Note: \ Binomial \ Series \ = \ (1+x)^{k} \ = \ 1+kx+\frac{k(k-1)x^{2}}{2!} \ + \ \frac{k(k-1)(k-2)x^{3}}{3!} \ + \ ...\)
\(\displaystyle Interval \ of \ Convergence \ is \ -1<x<1, \ and \ binomial \ series \ is \ valid \ for \ noninteger \ values \ of \ k.\)
