I would like to have an estimate for the infinite series $$ \sum_{k=B}^\infty \frac{A^k}{k!}, $$ where $A$ is a large positive quantity and $B$ is just a little bit bigger than $A$, namely, $B = A + C \sqrt A$ for some fixed large positive constant $C$. (In my application, $A$ and thus $B$ are increasing functions of some other variable, but $C$ really will stay fixed.)

I expect that the answer should look something like $$ ?\ \sum_{k=B}^\infty \frac{A^k}{k!} \ll e^{-C^2/2} \ \ ? $$ uniformly in $A$, $B$, and $C$. (Possibly there should even be an asymptotic formula.) It would be great to be able to just quote such an estimate "off the shelf". I've only been able to find such estimates when $B$ is substantially larger than $A$, such as $B > 5A$.

Does anyone know of a bound of this type in the literature? Many thanks.