Nested Squares Ratio

Let S₁be a square of side a. Another square S₂ is formed by joining the mid-points of the sides of S₁. The same process is applied to S₂ to form yet another square S₃, and so on. If A₁, A₂, A₃... are the areas and P₁, P₂, P₃... are the perimeters of S₁, S₂, S₃... respectively,

then the ratio $\frac{P_{1} + P_{2} + P_{3} + . . . . .}{A_{1} + A_{2} + A_{3} + . . .}$ equals:

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