I have the following complicated expression $E(n)$ and I am interested in finding a simple expression $S(n)$ such that $S(n) \leq E(n)$ for all sufficiently large $n$. Does anyone know how one can do this in maple?
$E(n) = $ $ 1/8\,\Biggl( 2\,\sqrt {2\,{n}^{2}-2\,n+1}\sqrt {2}n+\sqrt {2\,{n}^{2}- 2\,n+1}n\sqrt {{\frac {{n}^{2}+1}{{n}^{2}}}}-\sqrt {2\,{n}^{2}-2\,n+1} $ $ -\Biggl(18\,{n}^{4}-18\,{n}^{3}+13\,{n}^{2}+8\,\sqrt {2}{n}^{4}\sqrt { {\frac {{n}^{2}+1}{{n}^{2}}}}-8\,\sqrt {2}{n}^{3}\sqrt {{\frac {{n}^{2 }+1}{{n}^{2}}}}+4\,\sqrt {2}{n}^{2}\sqrt {{\frac {{n}^{2}+1}{{n}^{2}}} }-8\,\sqrt {2}{n}^{3}+ $ $8\,\sqrt {2}{n}^{2}-4\,\sqrt {2}n-4\,n+2-4\,{n}^ {3}\sqrt {{\frac {{n}^{2}+1}{{n}^{2}}}}+ $ $ 4\,{n}^{2}\sqrt {{\frac {{n}^{ 2}+1}{{n}^{2}}}}-2\,n\sqrt {{\frac {{n}^{2}+1}{{n}^{2}}}}+16\,\sqrt {2 \,{n}^{2}-2\,n+1}-16\,\sqrt {2\,{n}^{2}-2\,n+1}n\sqrt {{\frac {{n}^{2} +1}{{n}^{2}}}}\Biggr)^{-1/2}\Biggr) $ $ {\frac {1}{\sqrt {2\,{n}^{2}-2\,n+1}}} \left( -1+n\sqrt {{\frac {{n}^{2}+1}{{n}^{2}}}} \right) ^{-1} $