\(A=\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{8}-\frac{1}{16}\right)+\left(\frac{1}{32}-\frac{1}{64}\right)\)
\(A=\frac{1}{4}+\frac{1}{16}+\frac{1}{64}=\frac{21}{64}<\frac{21}{63}\)
\(\Rightarrow A<\frac{21}{63}=\frac{1}{3}\)
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