Đáp án đúng: A
Giải chi tiết:\(C = \frac{2}{{1.7}} + \frac{2}{{7.13}} + \frac{2}{{13.19}} + \ldots + \frac{2}{{2013.2019}}\)
Xét từng phân số ta thấy: Hiệu 2 thừa số ở mẫu bằng \(6\) \( \Rightarrow \) Nhân cả 2 vế của biểu thức với \(3\).
\(\begin{array}{l} \Rightarrow 3C = 3 \cdot \left( {\frac{2}{{1.7}} + \frac{2}{{7.13}} + \frac{2}{{13.19}} + \ldots + \frac{2}{{2013.2019}}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{6}{{1.7}} + \frac{6}{{7.13}} + \frac{6}{{13.19}} + \ldots + \frac{6}{{2013.2019}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{1}{1} - \frac{1}{7}} \right) + \left( {\frac{1}{7} - \frac{1}{{13}}} \right) + \left( {\frac{1}{{13}} - \frac{1}{{19}}} \right) + \ldots + \left( {\frac{1}{{2013}} - \frac{1}{{2019}}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{1} - \frac{1}{7} + \frac{1}{7} - \frac{1}{{13}} + \frac{1}{{13}} - \frac{1}{{19}} + \ldots + \frac{1}{{2013}} - \frac{1}{{2019}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1 - \frac{1}{{2019}} = \frac{{2018}}{{2019}}\end{array}\)
\( \Rightarrow 3C = \frac{{2018}}{{2019}} \Rightarrow C = \frac{{2018}}{{2019}}:3 = \frac{{2018}}{{2019}} \cdot \frac{1}{3} = \frac{{2018}}{{6057}}\)
Chọn A.
