Có:
\(Y=1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2012}}\)
\(\Rightarrow2Y=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2011}}\)
Lấy 2Y-Y, ta được:
\(2Y-Y=\left(2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2011}}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2012}}\right)\)
\(2Y-Y=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2011}}-1-\dfrac{1}{2}-\dfrac{1}{2^2}-\dfrac{1}{2^3}-...-\dfrac{1}{2^{2012}}\)\(2Y-Y=2+\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2011}}-1-\dfrac{1}{2}-\dfrac{1}{2^2}-\dfrac{1}{2^3}-...-\dfrac{1}{2^{2011}}\right)-\dfrac{1}{2^{2012}}\)
\(2Y-Y=2+\left(-\dfrac{1}{2^{2012}}\right)=2-\dfrac{1}{2^{2012}}\)
Hay \(Y=2-\dfrac{1}{2^{2012}}=\dfrac{2^{2013}-1}{2^{2012}}\)
Chúc bạn học tốt!