The Basel problem asks: what is the exact value of 1 + 1/4 + 1/9 + 1/16 + ⋯? The series converges, but to what? Pietro Mengoli posed it in 1650. It stumped every mathematician for 84 years until Euler solved it in 1734 at age 28.
Partial sums approach π²/6 ≈ 1.6449 slowly. Euler proved the limit equals π²/6 in 1734, connecting analysis to geometry.
Euler's proof factored the Taylor series for sin(x)/x as an infinite product over its roots ±π, ±2π, ±3π… Comparing the x² coefficient of the product form to the Taylor coefficient gives Σ 1/n² = π²/6 directly. It is one of the most celebrated computations in mathematics, and the reason π appears here is not coincidence: circles and spheres have natural connections to integer sums through the Riemann zeta function.
Each term 1/n^2 decreases rapidly. Their sum converges to exactly pi^2/6 ~1.6449.
The result generalises: ζ(4) = π⁴/90, ζ(6) = π⁶/945, and all even zeta values are rational multiples of powers of π. The odd values ζ(3), ζ(5), ζ(7)… are far more mysterious. Apéry proved ζ(3) is irrational in 1978, but no closed form in terms of π is known.
The probability that two randomly chosen integers share no common factor (are coprime) is exactly 6/pi^2, the reciprocal of pi^2/6. This is approximately 60.8%. It connects the Basel problem directly to number theory and probability.