Infinite Series Visual Proof

midpoints =
  NestList[Mean[{{0, 0}, {1, 0}, #}] &, {1/2, Sqrt[3]/2}, 10];
iter[n_] :=
  Graphics[Join[{White, EdgeForm[Black]},
    Table[
     {Polygon[{{0, 0}, {1, 0}, midpoints[[i]]}],
      Polygon[{midpoints[[i]], midpoints[[i - 1]], {1, 0}}],
      Red,
      Polygon[{midpoints[[i]], midpoints[[i - 1]], {0, 0}}]}
     , {i, 2, n}]
    ]];
iter[1] =
  Graphics[{White, EdgeForm[Black],
    Polygon[{{0, 0}, {1, 0}, {1/2, Sqrt[3]/2}}]}];
Manipulate[iter[n], {n, 1, 10, 1}]