Gabriel's Staircase

Author

Eric W. Weisstein
July 9, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/GabrielsStaircase.nb.

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

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

As a single series

Sum[k (r^k),{k,∞}]

As a double series

Sum[(r^i),{i,k,∞}]

Sum[%,{k,∞}]

Sum[(r^i),{k,∞},{i,k,∞}]

Graphic

FullSimplify[Sum[r^k,{k,n-1}]]

FullSimplify[Sum[r^k,{k,n}]]

Sum[r^k,{k,∞}]

Plot

<<Graphics`Colors`

ClosedLine[{x0_,y0_},{x1_,y1_}]:=Line[{{x0,y0},{x1,y0},{x1,y1},{x0,y1},{x0,y0}}]

Gabriel1[r_,n_]:=Table[ClosedLine[{(r^k-r)/(r-1),0},{r(r^k-1)/(r-1),k}],{k,n}]

Gabriel2[r_,n_]:=
Table[ClosedLine[{(r^k-r)/(r-1),k-1},{r/(1-r),k}],{k,n}]

Show[Graphics[{
{Gabriel1[.8,10]},
{Red,Gabriel2[.8,10]}
},AspectRatio->Automatic]]

Show[GraphicsArray[Graphics[
#,
AspectRatio->Automatic,
PlotRange->{{0,3},{0,8}}
]&/@{Gabriel1[.7,10],Gabriel2[.7,8]}
]]