A new elementary proof of the formula $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}=\frac{π^2}{6}$
Jia Li
TL;DR
The Basel problem is addressed with a new elementary proof that is self-contained and grounded in linear-algebra, extending Papadimitriou's approach. By constructing a structured matrix family $A_n$ and its inverse $B_n$, the authors derive a determinant $D_n(\theta)$ with a recurrence and a closed form, from which the eigenvalues of $A_n$ follow and lead to sums of trigonometric functions, notably $\sum_{k=1}^n\cot^2(\cdot)$ and $\sum_{k=1}^n\cot^4(\cdot)$. These identities feed into a trigonometric inequality-based argument that yields the exact value $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, and additionally confirms $\sum_{n=1}^{\infty}\frac{1}{n^4}=\frac{\pi^4}{90}$. The work thus provides a concise, algebraically flavored route to the Basel problem with clear connections between eigenvalue theory and zeta values.
Abstract
In this article, we provide a new elementary proof of the Basel problem.