Does anyone recognise this Taylor series expansion of an exp-like function?

Question

Does anyone recognise this Taylor series expansion? It is similar to that of $\exp(x)$, but not quite:

$$ 1 - \frac{1}{2!}x + \frac{1}{3!}x^2 - \frac{1}{4!}x^3 + \frac{1}{5!}x^4 - \frac{1}{6!}x^5 + \ldots $$

Is this a well-known function? Thank you very much in advance.

Answer 1

\begin{align*} S(x)&=1-\dfrac{1}{2!}x+\dfrac{1}{3!}x^{2}-\cdots\\ &=\dfrac{1}{x}\left(x-\dfrac{1}{2!}x^{2}+\dfrac{1}{3!}x^{3}-\cdots\right)\\ &=-\dfrac{1}{x}\left(-x+\dfrac{1}{2!}x^{2}-\dfrac{1}{3!}x^{3}+\cdots\right)\\ &=-\dfrac{1}{x}(e^{-x}-1). \end{align*}

And define $S(0)=1$.