We present a proof given by Euler in his paper “De serierum determinatione
seu nova methodus inveniendi terminos generales serierum”  (E189:“On
the determination of series or a new method of finding the general terms
of series”) for Stirling’s formula. Euler’s proof uses his theory of difference
equations with constant coefficients. This theory outgrew from his ear-
lier considerations on inhomogeneous differential equations with constant
coefficients of finite order that he tried to extend to the case of infinite
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"Euler’s First Proof of Stirling’s Formula,"
Euleriana: 3(2), pp.147-155.
Available at: https://scholarlycommons.pacific.edu/euleriana/vol3/iss2/6