\(B=1\dfrac{1}{2}+1\dfrac{1}{6}+1\dfrac{1}{12}+1\dfrac{1}{20}+...+1\dfrac{1}{380}+1\dfrac{1}{420}\)
\(=\left(1+\dfrac{1}{1\times2}\right)+\left(1+\dfrac{1}{2\times3}\right)+\left(1+\dfrac{1}{3\times4}\right)+...+\left(1+\dfrac{1}{19\times20}\right)+\left(1+\dfrac{1}{20\times21}\right)\)
\(=\left(1+1+...+1\right)+\left(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+...+\dfrac{1}{19\times20}+\dfrac{1}{20\times21}\right)\)
\(=20+\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{19}-\dfrac{1}{20}+\dfrac{1}{20}-\dfrac{1}{21}\right)\)
\(=20+\left(1-\dfrac{1}{21}\right)=\dfrac{440}{21}\)
