I learned something new today from visitor Sophos. He has an interesting geometric proof of the series 1/4 + 1/4^2 + … = 1/3. You can read it here. I think it is brilliant, and it reminds me of Tetris, an 80s arcade game where you try to score as high as possible by twisting and turning different edged geometrical objects to get a line without gaps. Think of a rotated L-shape that fits to the left of the L shape that is shaded, and think of another one where we “linearise” it and then use it as a base. This completes a square, of which the shaded L-shape makes up 1/3. The shaded L-shape can be thought of as a series of decreasing L-shapes by a factor of 1/4, so it equals to 1/4 + 1/4^2 + 1/4^3 + … . This completes the proof.
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