\(A=\dfrac{1}{2^1}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{100}}\)
\(\Rightarrow2A=1+\dfrac{1}{2^1}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}\)
\(\Rightarrow A=2A-A=1+\dfrac{1}{2^1}+...+\dfrac{1}{2^{99}}-\dfrac{1}{2^1}-\dfrac{1}{2^2}-...-\dfrac{1}{2^{100}}=1-\dfrac{1}{2^{100}}\)
\(=\dfrac{2^{100}-1}{2^{100}}=B\)