Visualizing a geometric series

Question

I attempted to graph the representation of the geometric series, everything works fine except when I increase "r" to a certain value(see figure 1 and 2 below), the horizontal lines begin to cross the diagonal line, therefore, making my model imperfect. Please help me to locate the bug mathematically.

Figure 1

Figure 2

Manipulate[Show[Graphics[{Red, Line[{{0, 0}, {0, a}}]}],
  (*the base square with side = a*)

  Graphics[{Red, Line[{{a, 0}, {a, a}}]}], 
  Graphics[{Red, Line[{{0, a}, {a, a}}]}], 
  Graphics[{Black, Line[{{0, 0}, {a, 0}}]}], 
  Graphics[{Black, Line[{{0, 0}, {a, 0}}]}],
  (*the vertical line*)

  Graphics[{Green, Line[{{a, 0}, {a, Sum[a r^i, {i, 0, 10}]}}]}],
  (*The diagonal line*)

  Graphics[{Blue, Line[{{0, 0}, {a, Sum[a r^i, {i, 0, 10}]}}]}],
  (*The horizontal lines*)

  Table[Graphics[{Black, 
     Line[{{a - 
         a r^i, (Sum[a r^b, {b, 0, i}] - 
          a r^i)}, {a, (Sum[a r^b, {b, 0, i}] - a r^i)}}]}], {i, 0, 
    10}],
  AxesLabel -> {"x", "y"}, PlotRange -> {-1, 5}], {a, 1, 4}, {r, 0.1, 
  0.9, 0.1}]

Answer 1

So if I understand correctly something like:

Manipulate[
 Show[Graphics[{Red, 
    Line[{{0, 0}, {0, a}}]}],(*the base square with side=a*)
  Graphics[{Red, Line[{{a, 0}, {a, a}}]}], 
  Graphics[{Red, Line[{{0, a}, {a, a}}]}], 
  Graphics[{Black, Line[{{0, 0}, {a, 0}}]}], 
  Graphics[{Black, Line[{{0, 0}, {a, 0}}]}],(*the vertical line*)
  Graphics[{Green, 
    Line[{{a, 0}, {a, Sum[a r^i, {i, 0, 10}]}}]}],(*The diagonal line*)
  Graphics[{Blue, 
    Line[{{0, 0}, {a, 
       Sum[a r^i, {i, 0, 10}]}}]}],(*The horizontal lines*)
  Table[Graphics[{Black,
     Line[{{(Sum[a r^b, {b, 0, i}] - a r^i)/(Sum[a r^i, {i, 0, 10}]/
         a), (Sum[a r^b, {b, 0, i}] - 
          a r^i)}, {a, (Sum[a r^b, {b, 0, i}] - a r^i)}}]}], {i, 0, 
    10}], AxesLabel -> {"x", "y"}, PlotRange -> {-1, 8}], {a, 1, 
  4}, {r, 0.1, 0.9, 0.1}]

might help the black lines from crossing the blue line as r increases?

To elaborate a little the division is just based on the fact that we have a blue line from 0,0 to wherever. I just use the old line equation y=mx+c (we can drop the +c part) rearranged to y/m=x to give us the x coordinate of where a black line xcoord sits in relation to our blue line. It isn't brilliant (I'm sure there are better ways, plus there is probably a way more efficient way of writing what you are doing in Mathematica) but that is just quickly how I saw it :)