Apostol's Solution to the Basel problem

The Basel problem asks for the solution to the infinite series of the reciprocal squares: $$ \sum_{n=1}^\infty \frac 1 {n^2} = \, ? $$As many other things in mathematics, it was first solved by Euler, who found that the solution was, incredibly, the transcendental value: $$ \frac{\pi^2}{6} $$If you read through the linked Wikipedia page, you’ll see many (quite beautiful) proofs, but they tend to rely on some decently heavy mathematical machinery. I was going through some old maths notes I made, and stumbled across an elegant proof from Tom M. Apostol1 that I wanted to reproduce here. Start by rewriting the sum so each term can be expressed as a product of two simple integrals: $$ \begin{aligned} \sum_{n=1}^{\infty} \frac{1}{n^2} &= \sum_{n=0}^{\infty} \frac{1}{(n+1)^2} &&\text{(reindexing)} \\ &= \sum_{n=0}^{\infty} \left[\frac{x^{n+1}}{n+1}\right]_0^1\left[\frac{y^{n+1}}{n+1}\right]_0^1 \\ &= \sum_{n=0}^{\infty} \left(\int_0^1 x^n \, \mathrm{d}x\right)\left(\int_0^1 y^n \,…