\[\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{1}{2}\left(1+%
\frac{1!}{1\cdot 3}+\frac{2!}{3\cdot 5}+\frac{3!}{3\cdot 5\cdot 7}+\cdots%
\right).\]