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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