Ta có : 1/2 = 0,5
2/3 = 0,666...
=> 1/2 + 2/3 + ... + 99/100 = 0,5 + 0,666...+3/4 + ... + 99/100
= 1,1,6666... + 3/4 + ... +99/100 > 1
=> 1/2 + 2/3 + ... + 99/100 > 1
\(=\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{99}{100}\le1\)
\(=\frac{2-1}{2}+\frac{3-1}{3}+\frac{4-1}{4}+...+\frac{100-1}{100}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(=1-\frac{1}{100}\le1\)
\(\Rightarrow1-\frac{1}{100}\le1\)
1/2 + 2/3 + 3/4 + ... + 99/100 < 1
= 2/2 - 1/2 + 3/3 - 1/3 + 4/4 - 1/4 + ... + 100/100 - 1/100
= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/99 - 1/100
= 1 - 1/100 < 1 (đpcm)
