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.

--Henri

PS: I learned this at the Math for America Utah excellent summer workshop.

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