Today’s entry is a Theorem of the Day:

The Basel Problem:

\[ \sum_{k=1}^\infty \frac{1}{k^2} = 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\tau^2}{24} \]

Originally posed in the 1640s, the value of this series was unknown until 1734 when it was solved by Euler. Many beautiful proofs exist; for some examples and more information, see the full listing at Theorem of the Day: the Basel Problem.

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!

At least one third of the editorial board of this website does not approve of the use of $\tau$ in the equation above.

$\frac{\pi^2}{6}$ for life!