\(A=\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{100}{2^{100}}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{2^2}+\frac{2}{2^3}+\frac{3}{2^4}+...+\frac{99}{2^{100}}+\frac{100}{2^{101}}\)
\(\Rightarrow A-\frac{1}{2}A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}-\frac{1}{2^{101}}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}-\frac{1}{2^{101}}\)
\(\Rightarrow A=1+\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{99}}-\frac{1}{2^{100}}\)
\(\Rightarrow2A=2+1+\frac{1}{2}+...+\frac{1}{2^{98}}-\frac{1}{2^{99}}\)
\(\Rightarrow2A-A=2-\frac{1}{2^{99}}-\frac{1}{2^{99}}+\frac{1}{2^{100}}\)
\(\Rightarrow A=2-\left(\frac{1}{2^{98}}-\frac{1}{2^{100}}\right)< 2\)
