\(\frac{1}{99.97}-\frac{1}{97.95}-...-\frac{1}{3.1}\)
\(=\frac{1}{99.97}-\left(\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{95.97}\right)\)
\(=\frac{1}{2}.\frac{2}{97.99}-\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{95.97}\right)\)
\(=\frac{1}{2}.\left[\frac{2}{97.99}-\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{95.97}\right)\right]\)
\(=\frac{1}{2}.\left[\frac{1}{97}-\frac{1}{99}-\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{95}-\frac{1}{97}\right)\right]\)
\(=\frac{1}{2}.\left[\frac{1}{97}-\frac{1}{99}-\left(1-\frac{1}{97}\right)\right]\)
\(=\frac{1}{2}.\left(\frac{1}{97}-\frac{1}{99}-\frac{98}{97}\right)\)
\(=\frac{1}{2}.\left(-1-\frac{1}{99}\right)\)
\(=\frac{1}{2}.\frac{-100}{99}\)
\(=-\frac{50}{99}\)
