Ta có : \(\frac{1}{2.2}< \frac{1}{1.2}\)
\(\frac{1}{3.3}< \frac{1}{2.3}\)
\(\frac{1}{4.4}< \frac{1}{3.4}\)
...................
\(\frac{1}{100.100}< \frac{1}{99.100}\)
Suy Ra : \(\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+......+\frac{1}{100.100}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+......+\frac{1}{99.100}\)
\(\frac{1}{2.2}+\frac{1}{3.3}+.....+\frac{1}{100.100}< 1-\frac{1}{100}=\frac{99}{100}< 1\)
Ta có : \(\frac{1}{2.2}\)\(< \frac{1}{1.2}\)
\(\frac{1}{3.3}\)\(< \frac{1}{2.3}\)
\(\frac{1}{4.4}\)\(< \frac{1}{3.4}\)
...... .... ......
\(\frac{1}{100.100}\)\(< \frac{1}{99.100}\)
\(\Rightarrow\)\(\frac{1}{2.2}\)+ \(\frac{1}{3.3}\)+ \(\frac{1}{4.4}\)+ ..... + \(\frac{1}{100.100}\)< \(\frac{1}{1.2}\)+ \(\frac{1}{2.3}\)+ \(\frac{1}{3.4}\)+ ..... + \(\frac{1}{99.100}\)
\(\frac{1}{2.2}\)+ \(\frac{1}{3.3}\)+ .... + \(\frac{1}{100.100}\)< \(1-\frac{1}{100}=\frac{99}{100}< 1\)
1/2.2 < 1/1.2
1/3.3 < 1/2.3
1/4.4 < 1/3.4
1/100.100 < 1/ 99.100
Nên 1/2.2 + 1/3.3 +1/4.4 + .... +1/100.100 < 1/1.1 +1/2.3+1/3.4 +......+ 1/99.100
1/2.2 + 1/3.3+.... 1/100.100 < 1 - 1/100 = 99/100 < 1
ta còn có 1 cách làm ngắn gọn hơn
