On Euler summation of factorials which are alternating in sign (Q1901900)

The series \(1! - 2! + 3! - 4! + \cdots\) (1) is summable by the method \((B^*)\) to the number \[ S^* = \int^\infty_0{xe^{- x}\over 1+x} dx = 0.4036 \dots; \] cf. \textit{G. H. Hardy} [Divergent series (1949; Zbl 0032.05801)]. In this paper the summability of (1) is investigated from the viewpoint of a new summation method -- the so-called Euler iterated method, shortly EI-method, depending on a given sequence \((\ell_n)^\infty_{n = 0}\) of nonnegative integer parameters. The EI-summability of (1) depends on the sequence \((\ell_n)^\infty_{n = 0}\) of parameters, e.g. if the series \(\sum^\infty_{n = 0} 2^{- n} \ell_n\) converges, then (1) is summable to a finite sum, which is less than \(S^*\) in the case when the \(\ell_n\) are positive even integers. If \(\ell_n = 2^{n + 1}\) \((n = 0,1, \dots)\) then (1) is summable to \(- \infty\).