So sánh hai biểu thức \(A\) và \(B\) biết: \(A = \frac{1}{{1.4}} + \frac{2}{{4.10}} + \frac{3}{{10.19}} + \frac{4}{{19.31}}\) và \(B = \frac{2}{{1.5}} + \frac{3}{{5.11}} + \frac{4}{{11.19}} +

Ta có: \(A = \frac{1}{{1.4}} + \frac{2}{{4.10}} + \frac{3}{{10.19}} + \frac{4}{{19.31}}\)

\(3A = \frac{3}{{1.4}} + \frac{6}{{4.10}} + \frac{9}{{10.19}} + \frac{{12}}{{19.31}}\)

\(3A = 1 - \frac{1}{4} + \frac{1}{4} - \frac{1}{{10}} + \frac{1}{{10}} - \frac{1}{{19}} + \frac{1}{{19}} - \frac{1}{{31}}\)

\(3A = 1 - \frac{1}{{31}}\)

\(3A = \frac{{30}}{{31}}\)

Do đó, \(A = \frac{{10}}{{31}}\).

Ta có: \(B = \frac{2}{{1.5}} + \frac{3}{{5.11}} + \frac{4}{{11.19}} + \frac{5}{{19.29}} + \frac{6}{{29.41}}.\)

\(2B = \frac{4}{{1.5}} + \frac{6}{{5.11}} + \frac{8}{{11.19}} + \frac{{10}}{{19.29}} + \frac{{12}}{{29.41}}\)

\(2B = 1 - \frac{1}{5} + \frac{1}{5} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{19}} + \frac{1}{{19}} - \frac{1}{{29}} + \frac{1}{{29}} - \frac{1}{{41}}\)

\(2B = 1 - \frac{1}{{41}}\)

\(2B = \frac{{40}}{{41}}\)

Do đó, \(B = \frac{{20}}{{41}}\).

Ta có: \(\frac{{10}}{{31}} = \frac{{20}}{{62}} < \frac{{20}}{{41}}\), do đó \(A < B.\)

Vậy \(A < B.\)