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Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

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Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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Binary Squares

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Clickety Click and All the Sixes

Stage: 5 Challenge Level: Challenge Level:1

What can you say about the number $111111$?
 
Can you write $666666$ as a series?