Final answer:
The sum of the areas of all the circles can be calculated by adding the areas of each circle. Let's represent the series of squares with n (where n is a positive integer) to find the sum of this infinite series.
Step-by-step explanation:
When a circle is inscribed in a square, the diameter of the circle is equal to the length of the side of the square. Let's say the side of the square is a. So, the diameter of the circle is also a. The radius of the circle is half the diameter, which is a/2.
The area of a circle is given by the formula A = πr^2, where π is a constant approximately equal to 3.14159 and r is the radius. So, substituting the value of r, we get A = π(a/2)^2 = πa^2/4.
Since we have a series of squares with inscribed circles, the sum of the areas of all the circles can be calculated by adding the areas of each circle. Let's represent the series of squares with n (where n is a positive integer). So, the sum of the areas of the circles is given by the formula S = πa^2/4 + π(a/2)^2/4 + π(a/4)^2/4 + ... + π(a/2^n)^2/4.
The sum of this infinite series can be found using the formula for the sum of a geometric series, which is S = a/(1 - r), where a is the first term and r is the common ratio. In this case, the first term is πa^2/4 and the common ratio is 1/4. Substituting these values, we get S = (πa^2/4)/(1 - 1/4) = (πa^2/4)/(3/4) = πa^2/3.