SOLVED: Challenge Question 1

If you are the first to solve this problem I will give you complete set of the Richard Feynman Lectures on Physics (read up upon the Feynman Lectures on Physics).

Question:
Take a square of arbitrary side length and construct four new squares with a side length half of the original positioned in the corners a quarter of the way down and across the two sides creating the corner. This would look like:

Then in each of the new squares in the furthest corner from the first square create a new square with half the side length a quarter of the way down and across the two sides creating the corner. Repeat this step for each newly constructed square to it infinity. To help visualise this I have created another graphic:

The question is, what proportion of the total area of the squares are overlapping with another square?

Leave answers either in the comments box or contact me with the answer through our contact page or email me at rouge.ray@gmail.com.