\(S=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{40}-\frac{1}{43}+\frac{1}{43}-\frac{1}{46}\)
\(S=1-\frac{1}{46}\)
Đến đây ta suy được ra S<1
Ta có :
\(S=\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{40.43}+\frac{3}{43.46}\)
\(S=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{40}-\frac{1}{43}+\frac{1}{43}-\frac{1}{46}\)
\(S=1-\frac{1}{46}\)
\(S=\frac{45}{46}< 1\)
Vậy \(S< 1\)
S=\(\frac{1}{1}\)-\(\frac{1}{4}\)+\(\frac{1}{4}\)-\(\frac{1}{7}\)+......+\(\frac{1}{40}\)-\(\frac{1}{43}\)+\(\frac{1}{43}\)-\(\frac{1}{46}\)
S=1-\(\frac{1}{46}\)<1
S=\(\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot10}+........+\frac{3}{43\cdot46}\) S= \(\frac{1}{1}-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+...............+\frac{3}{43}-\frac{3}{46}\) S=\(1+\left(\frac{-1}{4}+\frac{1}{4}\right)+..............+\left(\frac{-3}{43}+\frac{3}{43}\right)-\frac{3}{46}\) S=\(1-\frac{3}{46}=\frac{43}{46}\)
