Tính giá trị biểu thức: \(A\, = \,\frac{1}{3} + \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}}\)

Tính giá trị biểu thức:

\(A\, = \,\,\frac{1}{3} + \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}}\)

\(2 \times A\,\, = \,\,\frac{2}{{1 \times 3}} + \frac{2}{{3 \times 5}} + \frac{2}{{5 \times 7}} + \frac{2}{{7 \times 9}} + \frac{2}{{9 \times 11}} + \frac{2}{{11 \times 13}}\)

\(2 \times A\, = \,\frac{{3 - 1}}{{1 \times 3}} + \frac{{5 - 3}}{{3 \times 5}} + \frac{{7 - 5}}{{5 \times 7}} + \frac{{9 - 7}}{{7 \times 9}} + \frac{{11 - 9}}{{9 \times 11}} + \frac{{13 - 11}}{{11 \times 13}}\)

\(2 \times A\, = \,1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{13}}\)

\(2 \times A\, = 1 - \frac{1}{{13}}\)

\(\begin{array}{l}2 \times A\, = \,\frac{{12}}{{13}}\\A\, = \,\frac{{12}}{{13}}:2\, = \,\frac{6}{{13}}\end{array}\)