S=\lim_{n\to{\infty}}S_{n}=\lim_{n\to{\infty}}{(}{1-\sqrt{2}+\sqrt{n+2}-\sqrt{n+1}}{)}=1-\sqrt{2}+\lim_{n\to{\infty}}\frac{1}{\sqrt{n+2}+\sqrt{n+1}}=1-\sqrt{2}