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

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.

• How about $$x-\frac{x^2}{2!}+\frac{x^3}{3!}-\cdots?$$ – Lord Shark the Unknown Mar 12 '18 at 18:08
• Thanks a lot Lord Shark the Unknown, looks like user284331 used this as a step in his derivation. – user36247 Mar 12 '18 at 18:25

And define $S(0)=1$.