Logarithm of Factorial/Historical Note

Historical Note on Logarithm of Factorial
The sequence of $a_1, a_2, \ldots$ was established by during the course of his attempt extend the factorial to the real numbers.

However, although he established that $\left({\dfrac 1 2}\right)! = \dfrac {\sqrt \pi} 2$, he was not able to prove that this sum defined $n!$ for a general non-integer $n$.