The Basel problem asks: what is the exact sum of 1 + 1/4 + 1/9 + 1/16 + ⋯? The pattern is 1/n² for every integer n. It converges (the sum doesn't grow without bound) but to what value? Pietro Mengoli posed the question in 1650. It stumped mathematicians for 90 years.
In 1734, Leonhard Euler, aged 28, proved the answer is π²/6. The appearance of π in a sum involving nothing but whole numbers was completely unexpected π was known from circles, not from the harmonic series. Euler's proof connected analysis, geometry, and number theory in a way nobody had anticipated.
Euler's strategy was brilliant: he factored the Taylor series sin(x)/x = 1 − x²/3! + x⁴/5! − ⋯ as an infinite product over its roots at x = ±π, ±2π, ±3π…, then compared coefficients to extract Σ1/n² = π²/6. It is one of the most celebrated computations in mathematics.