Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

A Swiss Sum

Age 16 to 18 Challenge Level

The Basel problem, dating back to the 17th century, asks for the sum of the infinite series
$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \cdots$$