$M=\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+...+\frac{1}{2915}+\frac{1}{3135}$
$⇒2M=\frac{2}{3×5}+\frac{2}{5×7}+\frac{2}{7×9}+...+\frac{2}{53×55}+\frac{2}{55×57}$
$⇒2M=\frac{5-3}{3×5}+\frac{7-5}{5×7}+\frac{9-7}{7×9}+...+\frac{55-53}{53×55}+\frac{57-55}{55×57}$
$⇒2M=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{53}-\frac{1}{55}+\frac{1}{55}-\frac{1}{57}$
$⇒2M=\frac{1}{3}-\frac{1}{57}$
$⇒2M=\frac{6}{19}$
$⇒M=\frac{3}{19}$.
