The Basel Problem — π in the Sum of Reciprocal Squares

What just happened?

You were adding successive reciprocal squares: $1,\ \tfrac14,\ \tfrac19,\ \tfrac1{16},\dots$ The successive terms shrink quadratically, so the partial sum very quickly stops growing appreciably and settles just below a certain ceiling. That ceiling is exactly $\pi^2/6$ — and that is the biggest surprise: the number $\pi$, tied to the circle, appears in a purely arithmetic sum.

The Basel problem:

The task is to find the exact sum of the series of reciprocal squares. It was posed by Pietro Mengoli in 1644, and for nearly 90 years the greatest mathematicians (among them the Bernoullis) tried to solve it, without success. The name comes from the Swiss city of Basel, home of the Bernoulli family who worked on it.

Euler's solution (1735):

The problem was finally solved by the 28-year-old Leonhard Euler, who proved an astonishing result: the sum equals exactly $\pi^2/6 \approx 1.6449$. This achievement brought him immediate fame. The result also launched the study of the Riemann zeta function: in its language $\zeta(2)=\sum_{n=1}^{\infty}\tfrac1{n^2}=\tfrac{\pi^2}{6}$ — one of the pillars of number theory.

Convergent, not divergent — why?

Compare with the harmonic series $1+\tfrac12+\tfrac13+\dots$, which is divergent (its sum tends to infinity). There the terms $1/n$ shrink too slowly. Here the terms $1/n^2$ shrink much faster — quadratically — and that is enough for the sum to be finite. The series $\sum 1/n^2$ is therefore convergent, and its limit is precisely $\pi^2/6$.

Why this matters on the exam:

This is the classic pair to remember: $\sum 1/n$ — divergent, $\sum 1/n^2$ — convergent. The rate at which the terms shrink decides whether the series converges. The condition $a_n\to 0$ alone is not enough — what matters is how fast they shrink.

Remember: the series $\sum_{n=1}^{\infty} 1/n^2$ is convergent, and its sum equals exactly $\pi^2/6 \approx 1.6449$. The surprising appearance of $\pi$ in the sum of reciprocal squares is one of the most beautiful results in mathematics (Euler, 1735).