Stirling's Theorem - A Convergent Version of Stirling's Formula

A Convergent Version of Stirling's Formula

Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did not give a convergent series.

Obtaining a convergent version of Stirling's formula entails evaluating

One way to do this is by means of a convergent series of inverted rising exponentials. If

then

where

where s(n, k) denotes the Stirling numbers of the first kind. From this we obtain a version of Stirling's series

\begin{align} \ln(\Gamma (z)) & = \left( z-\tfrac{1}{2}\right) \ln(z) -z + \tfrac{1}{2}\ln(2 \pi) + \frac{1}{12(z+1)} + \frac{1}{12(z+1)(z+2)} + \\ & \qquad \qquad + \frac{59}{360(z+1)(z+2)(z+3)} + \frac{29}{60(z+1)(z+2)(z+3)(z+4)} + \cdots \end{align}

which converges when Re(z) > 0.

Read more about this topic:  Stirling's Theorem

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