Ta có: \(H=\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\cdots+\frac{1}{2009\cdot2010}\)
\(=1-\frac12+\frac13-\frac14+\cdots+\frac{1}{2009}-\frac{1}{2010}\)
\(=1+\frac12+\frac13+\frac14+\cdots+\frac{1}{2009}+\frac{1}{2010}-2\left(\frac12+\frac14+\cdots+\frac{1}{2010}\right)\)
\(=1+\frac12+\cdots+\frac{1}{2010}-1-\frac12-\cdots-\frac{1}{1005}\)
\(=\frac{1}{1006}+\frac{1}{1007}+\cdots+\frac{1}{2010}\)
Ta có: \(L=\frac{1}{1006\cdot2010}+\frac{1}{1007\cdot2009}+\cdots+\frac{1}{2010\cdot1006}\)
\(=2\left(\frac{1}{1006\cdot2010}+\frac{1}{1007\cdot2009}+\cdots+\frac{1}{1507\cdot1509}\right)+\frac{1}{1508\cdot1508}\)
\(=\frac{2}{3016}\left(\frac{3016}{1006\cdot2010}+\frac{3016}{1007\cdot2009}+\cdots+\frac{3016}{1507\cdot1509}\right)+\frac{1}{1508\cdot1508}\)
\(=\frac{1}{1508}\left(\frac{1}{1006}+\frac{1}{2010}+\frac{1}{1007}+\frac{1}{2009}+\cdots+\frac{1}{1507}+\frac{1}{1509}\right)+\frac{1}{1508}\cdot\frac{1}{1508}\)
\(=\frac{1}{1508}\left(\frac{1}{1006}+\frac{1}{1007}+\cdots+\frac{1}{2010}\right)\)
Ta có: \(\frac{H}{L}\)
\(=1:\frac{1}{1508}=1508\)
=>H/L ∈Z
