Question

CMR \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{1317}-\frac{1}{1318}+\frac{1}{1319}\)     =      \(\frac{1}{660}+\frac{1}{661}+......+\frac{1}{1319}\)

Answer 1

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+........+\frac{1}{1317}-\frac{1}{1318}+\frac{1}{1319}\)

\(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+.......+\frac{1}{1319}-2.\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+.....+\frac{1}{1318}\right)\)

\(=1+\frac{1}{2}+\frac{1}{3}+........+\frac{1}{1319}-\left(1+\frac{1}{2}+\frac{1}{3}+.......+\frac{1}{658}+\frac{1}{659}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+........+\frac{1}{659}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+......+\frac{1}{659}\right)+\frac{1}{660}+\frac{1}{661}+......+\frac{1}{1319}\)

\(=\frac{1}{660}+\frac{1}{661}+.........+\frac{1}{1319}\)