\(\dfrac{1}{15}+\dfrac{1}{35}+\dfrac{1}{99}+\dfrac{1}{143}+\dfrac{1}{195}\\ =\dfrac{1}{15}+\dfrac{1}{35}+\dfrac{1}{63}+\dfrac{1}{99}+\dfrac{1}{143}+\dfrac{1}{195}-\dfrac{1}{63}\\ =\dfrac{1}{2}\left(\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}+\dfrac{2}{9.11}+\dfrac{2}{11.13}+\dfrac{2}{13.15}\right)-\dfrac{1}{63}\\ =\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{13}+\dfrac{1}{13}-\dfrac{1}{15}\right)-\dfrac{1}{63}\\ =\dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{15}\right)-\dfrac{1}{63}=\dfrac{1}{2}.\dfrac{4}{15}-\dfrac{1}{63}=\dfrac{2}{15}-\dfrac{1}{63}=\dfrac{37}{315}\)
