1/4 + 1/16 + 1/64 + 1/256 + · · ·
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High Quality Content by WIKIPEDIA articles! The series 1/4 + 1/16 + 1/64 + 1/256 + · · · lends itself to some particularly simple visual demonstrations because a square and a triangle both divide into four similar pieces, each of which contains 1/4 the area of the original. In the figure on the left, if the large square is taken to have area 1, then the largest black square has area (1/2)(1/2) = 1/4. Likewise, the second largest black square has area 1/16, and the third largest black square has area 1/64. The area taken up by all of the black squares together is therefore 1/4 + 1/16 + 1/64 + ·…mehr

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High Quality Content by WIKIPEDIA articles! The series 1/4 + 1/16 + 1/64 + 1/256 + · · · lends itself to some particularly simple visual demonstrations because a square and a triangle both divide into four similar pieces, each of which contains 1/4 the area of the original. In the figure on the left, if the large square is taken to have area 1, then the largest black square has area (1/2)(1/2) = 1/4. Likewise, the second largest black square has area 1/16, and the third largest black square has area 1/64. The area taken up by all of the black squares together is therefore 1/4 + 1/16 + 1/64 + · · ·, and this is also the area taken up by the gray squares and the white squares. Since these three areas cover the unit square, the figure demonstrates that.