\(\frac{8}{9}-\frac{1}{72}-\frac{1}{56}-\frac{1}{42}-\frac{1}{30}-\frac{1}{20}-\frac{1}{12}-\frac{1}{6}-\frac{1}{2}\)
\(=\frac{8}{1.9}-\frac{1}{8.9}-\frac{1}{7.8}-\frac{1}{5.6}-\frac{1}{4.5}-\frac{1}{3.4}-\frac{1}{2.3}-\frac{1}{1.2}\)
\(=1-\frac{1}{9}-\frac{1}{8}+\frac{1}{9}-\frac{1}{7}+\frac{1}{8}-\frac{1}{5}+\frac{1}{6}-\frac{1}{4}+\frac{1}{5}-\frac{1}{3}+\frac{1}{4}-\frac{1}{2}+\frac{1}{3}-1+\frac{1}{2}\)
\(=0\)
8/9 - 1/72 - 1/56 - ... - 1/6 - 1/2
= 8/9 - [1/8*9 + 1/7*8 + ... + 1/2*3 + 1/1*2]
= 8/9 - [1/8 - 1/9 + 1/7 - 1/8 + ...+ 1/2 - 1/3 + 1 - 1/2]
= 8/9 - [-1/9 + 1]
= 8/9 - 8/9
= 0
