$B = \frac{4}{{3\, \times \,7}}\,\, + \,\,\frac{4}{{7\, \times \,11}}\,\, + \,\,\frac{4}{{11\, \times \,15}}\,\, + \,\frac{4}{{15\, \times \,19}}\,\, + \,\,\frac{4}{{19\, \times \,23}}\,\, + \,\frac{4}{{23\, \times \,27}}$ $B = \frac{{7 - 3}}{{3 \times 7}} + \frac{{11 - 7}}{{7 \times 11}} + \frac{{15 - 11}}{{11 \times 15}} + \frac{{19 - 15}}{{15 \times 19}} + \frac{{23 - 19}}{{19 \times 23}} + \frac{{27 - 23}}{{23 \times 27}}$ $B = \frac{1}{3} - \frac{1}{7} + \frac{1}{7} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{15}} + \frac{1}{{15}} - \frac{1}{{19}} + \frac{1}{{19}} - \frac{1}{{23}} + \frac{1}{{23}} - \frac{1}{{27}}$ $B = \frac{1}{3} - \frac{1}{{27}} = \frac{8}{{27}}$