Notes on Euler's work on divergent factorial series and their associated continued fractions (Q981851)

Let \(f=1-1!x+2!x^2-3!x^3+\ldots \). Euler obtained a continued fraction for the formal series \(f\): \[ f=[1,x,x,2x,2x,3x,3x,\ldots,nx,nx,\ldots]. \] The authors present a proof of this and establish more general identities.