Tính giá trị biểu thức:

\[\frac{1}{2}\] + \[\frac{1}{6}\] + \[\frac{1}{{12}}\] + \[\frac{1}{{20}}\] + \[\frac{1}{{30}}\] + \[\frac{1}{{42}}\] + \[\frac{1}{{56}}\]

= \[\frac{1}{{1 \times 2}}\]+ \[\frac{1}{{2 \times 3}}\] + \[\frac{1}{{3 \times 4}}\] + \[\frac{1}{{4 \times 5}}\] + \[\frac{1}{{5 \times 6}}\] + \[\frac{1}{{6 \times 7}}\] + \[\frac{1}{{7 \times 8}}\]

= 1 – \[\frac{1}{2}\] + \[\frac{1}{2}\] – \[\frac{1}{3}\] + \[\frac{1}{3}\] – \[\frac{1}{4}\] + \[\frac{1}{4}\] – \[\frac{1}{5}\] + \[\frac{1}{5}\] – \[\frac{1}{6}\] + \[\frac{1}{6}\] – \[\frac{1}{7}\] + \[\frac{1}{7}\] – \[\frac{1}{8}\]

= 1 + (\[\frac{1}{2}\] – \[\frac{1}{2}\]) + (\[\frac{1}{3}\] – \[\frac{1}{3}\] ) + (\[\frac{1}{4}\]– \[\frac{1}{4}\]) + (\[\frac{1}{5}\] – \[\frac{1}{5}\]) + (\[\frac{1}{6}\] – \[\frac{1}{6}\]) + (\[\frac{1}{7}\] – \[\frac{1}{7}\]) – \[\frac{1}{8}\]

= 1 – \[\frac{1}{8}\] = \[\frac{7}{8}\]