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1 Sums of reciprocals
Euler found the sum of the reciprocals of the squares in 1734 [3]. Theorem 1 states it; a modern account is in [1], and the story of the problem in [2]. The proof in Section 2 rests on the product formula (1).
Theorem 1 (Euler [3])..
2 The proof
The sine is the product
Comparing the coefficients ofon both sides gives the theorem; the last step is written out below. Why the problem has the name it has is in the last section.
The coefficient ofison the left of (1) andon the right.
3 The name
Basel was the home of the Bernoullis, who made the problem known [2]. More on the Basel problem at https://en.wikipedia.org; displayed formulas in general on the page Displays.
References
[1] M. Aigner and G. M. Ziegler. Proofs from THE BOOK, 6th edition. Springer, 2018.
[2] W. Dunham. Euler: The Master of Us All. MAA, 1999.
[3] L. Euler. De summis serierum reciprocarum. Commentarii academiae scientiarum Petropolitanae 7 (1740), 123–134.
\section{Sums of reciprocals}\label{sec:sums}Euler found the sum of the reciprocals of the squares in
1734~\cite{euler1740}. \autoref{thm:basel} states it; a modern account is
in~\cite{aigner2018}, and the story of the problem in~\cite{dunham1999}.
The proof in Section~\ref{sec:proof} rests on the product
formula~\eqref{eq:product}.
\begin{theorem}[Euler~\cite{euler1740}]\label{thm:basel}$\displaystyle\sum_{n \ge1} \frac{1}{n^2} =\frac{\pi^2}{6}$.
\end{theorem}\section{The proof}\label{sec:proof}The sine is the product
\begin{equation}\label{eq:product}\frac{\sin x}{x} = \prod_{n \ge 1}\Bigl(1 - \frac{x^2}{n^2 \pi^2}\Bigr).
\end{equation}Comparing the coefficients of $x^2$ on both sides gives the theorem; the
\hyperlink{last-step}{last step} is written out below. Why the problem has
the name it has is in \hyperref[sec:name]{the last section}.
\hypertarget{last-step}{The coefficient} of $x^2$ is $-1/6$ on the left
of~\eqref{eq:product} and $-\sum_{n \ge1} 1/(n^2\pi^2)$ on the right.
\section{The name}\label{sec:name}Basel was the home of the Bernoullis, who made the problem
known~\cite{dunham1999}. More on
\href{https://en.wikipedia.org/wiki/Basel_problem}{the Basel problem} at
\url{https://en.wikipedia.org}; displayed formulas in general on the page
\href{../displays/}{Displays}.
\begin{thebibliography}{9}\bibitem{aigner2018} M.~Aigner and G.~M.~Ziegler.
\emph{Proofs from THE BOOK}, 6th edition. Springer, 2018.
\bibitem{dunham1999} W.~Dunham.
\emph{Euler: The Master of Us All}. MAA, 1999.
\bibitem{euler1740} L.~Euler. De summis serierum reciprocarum.
\emph{Commentarii academiae scientiarum Petropolitanae} 7 (1740), 123--134.
\end{thebibliography}
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