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.



Estimate for tail of power series of exponential function?
Let's instead consider the sum


$$ \sum\_{k = A + C \sqrt{A}}^\infty {e^{-A} A^k \over k!} $$
which of course differs from yours just by a factor of $e^{-A}$.


Then this sum is the probability that a Poisson random variable of mean $A$ is at least $A + C\sqrt{A}$.


A Poisson with mean $A$ has standard deviation $\sqrt{A}$, and as $A \to \infty$ the Poissons become asymptotically normal. So we have


$$ \sum\_{k = A + C \sqrt{A}}^\infty {e^{-A} A^k \over k!} \to \Phi(C) $$


as $A \to \infty$, where $\Phi$ is the CDF of the standard normal. So your sum is asymptotic to $e^A \Phi(C)$.


Alternatively, if you'd like an explicit inequality, your sum can be bounded above by the geometric series with first term $A^B/B!$ and common term ratio $A/B$. Therefore, your sum is less than
$$ {A^B \over B!} {1 \over 1-A/B} $$
and this can be rewritten as
$$ {A^B \over B!} \left( 1 + {\sqrt{A} \over C} \right) $$
The product $A^B/B!$ is, as $A \to \infty$ with $B = A + C \sqrt{A}$,
$$ {1 \over \sqrt{2\pi}} e^{-C^2/2} A^{-1/2} e^A (1+o(1))$$
by Stirling's formula. In the factor $1 + \sqrt{A}/C$ we can neglect $1$ as $A \to \infty$, so we get that 


$$ \sum\_{k = A+C\sqrt{A}}^\infty {e^{-A} A^k \over k!} \le {1 \over \sqrt{2\pi}} C e^{-C^2/2} e^A (1 + o(1)) $$


By, say, [the double inequality (26) here](http://mathworld.wolfram.com/StirlingsApproximation.html) it should be possible to get explicit bounds.


