Ta có:
\(A=\dfrac{1}{4}+\dfrac{1}{4^2}+\dfrac{1}{4^3}+...+\dfrac{1}{4^{1000}}\)
\(4A=4\cdot\left(\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{1000}}\right)\)
\(4A=1+\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{999}}\)
\(4A-A=1+\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{999}}-\dfrac{1}{4}-\dfrac{1}{4^2}-...-\dfrac{1}{4^{1000}}\)
\(3A=1-\dfrac{1}{4^{1000}}\)
\(A=\dfrac{1-\dfrac{1}{4^{1000}}}{3}\)
Mà: \(1-\dfrac{1}{4^{1000}}< 1\)
\(\Rightarrow\dfrac{1-\dfrac{1}{4^{1000}}}{3}< \dfrac{1}{3}\)
\(\Rightarrow A< \dfrac{1}{3}\)
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