Question

Stacking boxes onto shelves at an Amazon warehouse is exhausting work. This problem may help you sympathize with those who do it. In the usual good news/bad news dichotomy, the good news is that the boxes here are two-dimensional, so they have no mass. The bad news is that there are infinitely many of them. ... For each integer n≥1, let aₙ =1/n² be the area of a square of side length 1/n. (Call this "box number n ".) Consider the task of piling all those boxes onto a horizontal shelf. Like physical boxes, one box can rest on top of another, but they can't occupy the same space.

Answer 1

The total area occupied by all the boxes when stacked on the horizontal shelf is ∑(1/n²) as n ranges from 1 to infinity.

Step-by-step explanation:

Each box corresponds to a square with side length 1/n, and its area is given by aₙ = 1/n². To find the total area occupied by all the boxes, we sum up the areas of all the individual boxes using the series notation ∑(1/n²) from n=1 to infinity.

The series ∑(1/n²) is known as the Basel problem, and its convergence was famously proven by Euler in the 18th century. Euler showed that the sum of the reciprocals of the squares of natural numbers converges to a finite value, specifically, π²/6. Mathematically, ∑(1/n²) = π²/6.

Therefore, the total area occupied by all the boxes when stacked on the horizontal shelf is π²/6. This means that even though there are infinitely many boxes, their combined area is a finite value, providing a surprising result and insight into the convergence of certain mathematical series.

Question:

What is the total area occupied by all the boxes when they are stacked on the horizontal shelf?