The summaration of series is

$\frac{1}{1.2}+\frac{1}{2.3}+ \dots +\frac{1}{99.100}\\=\frac{2-1}{1.2}+\frac{3-2}{2.3}+ \dots +\frac{100-99}{99.100}\\=\frac{2}{1.2} + \frac{-1}{1.2}+\frac{3}{2.3}+\frac{-2}{2.3}+ \dots +\frac{100}{99.100}+\frac{-99}{99.100}\\=1-\frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4}+ \dots +\frac{1}{99} - \frac{1}{100}\\=1-\frac{1}{100} = 0.99$

$\frac{1}{1.2}+\frac{1}{2.3}+ \dots +\frac{1}{99.100}\\=\frac{2-1}{1.2}+\frac{3-2}{2.3}+ \dots +\frac{100-99}{99.100}\\=\frac{2}{1.2} + \frac{-1}{1.2}+\frac{3}{2.3}+\frac{-2}{2.3}+ \dots +\frac{100}{99.100}+\frac{-99}{99.100}\\=1-\frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4}+ \dots +\frac{1}{99} - \frac{1}{100}\\=1-\frac{1}{100} = 0.99$