Cho P = 1/4.8 + 1/8.12 + . . . + 1/( 4 x − 4 ) .4 x (với x ∈ N , x ≥ 2 ). Chứng minh rằng P < 1/16 .

Ta có \(P = \frac{1}{{4.8}} + \frac{1}{{8.12}} + ... + \frac{1}{{\left( {4x - 4} \right).4x}}\)

\(\begin{array}{l} = \frac{1}{4}\left( {\frac{1}{4} - \frac{1}{8} + \frac{1}{8} - \frac{1}{{12}} + ... + \frac{1}{{\left( {4x - 4} \right)}} - \frac{1}{{4x}}} \right)\\ = \frac{1}{4}\left( {\frac{1}{4} - \frac{1}{{4x}}} \right)\\ = \frac{1}{{16}} - \frac{1}{{16x}}\end{array}\)

Ta thấy \(\frac{1}{{16x}} > 0\) nên \(\frac{1}{{16}} - \frac{1}{{16x}} < \frac{1}{{16}}\)

Vậy với \(x \ge 2\) thì \(P < \frac{1}{{16}}\).