ta có Đặt \(A=\frac{2008^{2008}+1}{2008^{2009}+1}\)
\(B=\frac{2008^{2007}+1}{2008^{2008}+1}\)
Xét A trước ta có
\(2008A=\frac{2008\left(2008^{2008}+1\right)}{2008^{2009}+1}\)\(2008A=\frac{2008^{2009}+2008}{2008^{2009}+1}\)
\(2008A=\frac{2008^{2009}+1+2007}{2008^{2009}+1}\)suy ra \(2008A=1+\frac{2007}{2008^{2009}+1}\)
Xét B ta có
\(2008B=\frac{2008.\left(2008^{2007}+1\right)}{2008^{2008}+1}\)suy ra \(2008B=\frac{2008^{2008}+2008}{2008^{2008}+1}\)
\(2008B=\frac{2008^{2008}+1+2007}{2008^{2008}+1}\)suy ra \(2008B=1+\frac{2007}{2008^{2008}+1}\)
VÌ \(1+\frac{2007}{2008^{2009}+1}
Đặt \(a=2008^{2007};\)
\(A=\frac{2008^{2008}+1}{2008^{2009}+1}=\frac{2008a+1}{2008^2.a+1};\text{ }B=\frac{2008^{2007}+1}{2008^{2008}+1}=\frac{a+1}{2008a+1}\)
Quy đồng mẫu ta có:
\(A=\frac{\left(2008a+1\right)\left(2008a+1\right)}{\left(2008^2a+1\right)\left(2008a+1\right)}=\frac{2008^2a^2+2.2008a+1}{\left(2008^2a+1\right)\left(2008a+1\right)}\)
\(B=\frac{\left(a+1\right)\left(2008^2a+1\right)}{\left(2008a+1\right)\left(2008^2a+1\right)}=\frac{2008^2a^2+\left(2008^2+1\right)a+1}{\left(2008a+1\right)\left(2008^2a+1\right)}\)
So sánh ở tử ta thấy \(2.2008
