Weird large K symbol

Solution 1:

It is the notation for a continued fraction. In general: $$b_0 + \underset{k=1}{\overset{\infty}{\large{\mathrm K}}} \left(\frac{a_k}{b_k}\right)=b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}}$$ Therefore, the continued fraction representation you have written above for $\pi$ is: $$\pi=3 + \underset{k=1}{\overset{\infty}{\large{\mathrm K}}} \frac{(2k-1)^2}{6}=3 + \cfrac{1^2}{6 + \cfrac{3^2}{6 + \cfrac{5^2}{6 + \cfrac{7^2}{6 + \ddots\,}}}}$$ A proof of this result can be found on pages 399-401 of this document by Paul Loya.

Solution 2:

The given symbol is sometimes used for an infinite continued fraction, it seems to have been designed by Carl Friedrich Gauss.