Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)...\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$ I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer.
I am interested to find the bounds on the value it can take or an approximation for it. Since $$0<\frac{a..\left( a+n-1 \right)}{\left( a+b \right)...\left( a+b+n-1 \right)}<1, $$ 
I was thinking that ${{e}^{z}}$ would be an upper bound. Is that right?
I am not sure about lower bound, can this go to $-\infty$?
$$ _{1}{{F}_{1}}\left( a,a+b,z \right)=\sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)...\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}\le {{e}^{z}}$$
Thanks!

You may be interested in the asymptotic formula,
$$
{}_1F_1(a,a+b,z) = \frac{\Gamma(a+b)}{\Gamma(b)} (-z)^{-a} + O(z^{-a-1})
$$
as $\operatorname{Re} z \to -\infty$ (see, e.g., [1]).
Note, in particular, that it is not true that ${}_1F_1(a,a+b,x) \leq e^x$ for $x \in \mathbb{R}$ large and negative.
[1] Bateman Manuscript Project, Higher Transcendental Functions.  Vol. 1, p. 248,255,278. [pdf]
