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Finite volume, infinite surface area of infinitely many cubes with decreasing edge lengths
Published on October 21, 2024
Calculus
The total volume of infinitely many cubes with side lengths decreasing according to the sequence 1, 1/√2, 1/√3, ... is finite, while the total surface area is infinite.
Question
Answer
Let’s analyze the volumes and surface areas separately.
Volumes
- The side lengths of the cubes form a sequence:
- The volume of each cube is the side length cubed. So, the volumes form a sequence:
which can be written as - This is a sequence where the nth term is given by
or . - The sum of the volumes of all the cubes is then given by the infinite series:
- This is a p-series where p = 3/2, which is greater than 1. A p-series converges if p > 1. Therefore, the sum of the volumes is finite.
Surface Areas
- Each cube has 6 faces.
- The area of one face of the nth cube is the side length squared:
. - The total surface area of the nth cube is then
. - The sum of the surface areas of all the cubes is given by the infinite series:
- This is a constant multiple of the harmonic series, which is known to diverge. Therefore, the sum of the surface areas is infinite.
Conclusion
We have proven that the total volume of the cubes is finite, while their total outer surface area is infinite. This means you could theoretically fill all the cubes with a finite amount of liquid, but you would need an infinite amount of paint to cover their surfaces.