Square Root Inequality

Author

Eric W. Weisstein
March 1, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/SquareRootInequality.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SquareRootInequality.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Inequality

ineq={2(Sqrt[n+1])-2(Sqrt[n]),(1/(Sqrt[n])),2(Sqrt[n])-2(Sqrt[n-1])};

Reduce[Less@@ineq,n,Reals]

Limits

Limit[Subtract@@@Partition[ineq,2,1],n->∞]

Plot

Plot[{2(Sqrt[n+1])-2(Sqrt[n]),(1/(Sqrt[n])),2(Sqrt[n])-2(Sqrt[n-1])},{n,1,20},PlotStyle->{Red,Black,Blue},AxesLabel->TraditionalForm/@{n,StyleForm[#1,FontColor->#2]&@@@Transpose[{ineq,{Red,Black,Blue}}]},
PlotRange->{{1,20},{0,1}},AxesOrigin->{1,0}]