Is it possible to fit smallers squares into a unit square with the following areas: sum from i=2 to infinite(1/i^2),
and if its possible how could i do that? I know that the sum of those squares are less than one but i didnt manage to find such an algorithm
(for how i should place the smaller squares) that would provide us the anwser. Any idea?
Let \(\displaystyle \ S_2 \ \) be the square with side length of 1/2.
And in general, let \(\displaystyle \ S_n \ \) be the square with side length of 1/n, for n a positive integer.
Place these squares flush against one of the interior sides of the unit square.
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\(\displaystyle S_2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S_3\)
\(\displaystyle S_4S_5 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S_7S_6\)
\(\displaystyle S_8S_9S_{10}S_{11}\ \ \ \ \ S_{15}S_{14}S_{13}S_{12}\)
\(\displaystyle \cdot\cdot\cdot \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot\cdot\cdot \)
The side lengths along the left-hand side form an infinite geometric sequence. Its corresponding series adds up to 1,
the side length of the unit square.
The side lengths along the right-hand side form an infinite geometric sequence. Its corresponding series adds up to 2/3,
less than the side length of the unit square.