jmc

Since $39=2\times 18+3$, the $39$th digit past the decimal point is the same as the $3$rd digit past the decimal point. To find this, we can directly divide: $$\begin{array}{ccccc}\text{multicolumn}2r0& .0& 5& 2& \\ \text{cline}2-519& 1& .0& 0& 0\\ \text{multicolumn}2r& 9& 5& \downarrow & \\ \text{cline}2-4\text{multicolumn}2r& & 5& 0& \\ \text{multicolumn}2r& & 3& 8& \\ \text{cline}4-5\text{multicolumn}2r& & 1& 2& \end{array}$$ Thus, the answer is $\boxed{2}$.