Start with a right triangle ∆ABC, with hypotenuse AB = 1, and AC = b.Where does the sum of the infinite geometric series b + b^2 + b^3 + ... appear in the figure?
Draw a parallel to AB through C.
Draw a circle centered at C, with radius b.
Label the intersection of the line and the circle D.
Drop a perpendicular from D to line AC, intersecting it at E.
What is the length of CE?
Draw the line BD.
Draw a parallel to CD through E, intersecting line BD at F.
Drop a perpendicular from F to line AC, intersecting it at G.
What is the length of EG?
Continue in this manner: parallel, perpendicular, parallel, perpendicular, and so on.
PS: I learned this at the Math for America Utah excellent summer workshop.