[tex]\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{9900} \\ \\ =\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100} \\ \\ =\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100} \\ \\ =1-(\frac{1}{2}-\frac{1}{2})-(\frac{1}{3}-\frac{1}{3})-...-(\frac{1}{99}-\frac{1}{99})-\frac{1}{100} \\ \\ 1-\frac{1}{100} \\ \\ =\frac{99}{100}[/tex]
