Ta có:A= \(\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{37.38}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{37}-\frac{1}{38}\)
\(=\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{37}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{38}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{37}+\frac{1}{38}\right)-2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{38}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{38}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{19}\right)\)
\(=\frac{1}{20}+\frac{1}{21}+...+\frac{1}{38}\)
B=\(\frac{1}{58}\left(\frac{58}{20.38}+\frac{58}{21.37}+...+\frac{58.}{38.20}\right)\)
=\(\frac{1}{58}\left(\frac{20+38}{20.38}+\frac{21+37}{21+37}+...+\frac{38+20}{38.20}\right)\)
\(=\frac{1}{58}\left(\frac{1}{20}+\frac{1}{38}+\frac{1}{21}+\frac{1}{37}+...+\frac{1}{38}+\frac{1}{20}\right)\)
\(=\frac{1}{58}.2\left(\frac{1}{20}+\frac{1}{21}+...+\frac{1}{38}\right)=\frac{A}{29}\)
=> \(\frac{A}{B}=29\)
