What is a good (strict) upperbound for the probability for having at least $c*N$ (where $c \leq 1$) successes in $N$ trials? I have $N$ trials, where I succeed each trial with probability $p$.  If we ask for the probability of having at least $q$ successes, using the binomial distribution we can write this as the sum:
$P[\geq q$ successes$]=\sum^{N}_{i=q}$${N} \choose {i}$$p^i*(1-p)^{N-i} = $${N} \choose {q}$ $[(1-p)^{N-q} p^q]$ $[$$_2F_1(1,q-N,1+q,\frac{p}{p-1})]$ (where $_2F_1$ is the hypergeometric function)
Which is a bit unpleasant looking.  Now imagine that I set $q = Floor[cN]$ (or $q = Ceiling[cN]$ if that makes any difference), where $c \leq 1$ is a constant.
What is a good, simple, and strict upperbound for $P[\geq q$ successes$]$?  Should I be concerned about the normal approximation to the binomial distribution with small values of $N$?
Notice: I made a typo earlier when calculating an exact value for $P[\geq q$ successes$]$.  The new result should be correct, but hardly seems to simplify things.
When, if ever, is it fair to say that the probability decreases at least exponentially with fixed $p$ and increasing $N$?

The regime $c\gt p$ fixed, $N\to\infty$, is called a large deviations event since, if each random variable $X_N$ is binomial $(N,p)$, $P[X_N\geqslant cN]\to0$. The general idea in this situation is that
$$
P[X_N\geqslant cN]\leqslant\mathrm e^{-N\cdot I(c)},\qquad P[X_N\geqslant cN]=\mathrm e^{-N\cdot I(c)+o(N)},
$$
for some $I(c)$ which is obtained by optimizing the exponential inequality bounding $P[X_N\geqslant cN]$. More concretely, note that for every $t\gt0$,
$$
P[X_N\geqslant cN]\leqslant\mathrm e^{-tcN}E[\mathrm e^{tX_N}]=\mathrm e^{-tcN}(1-p+p\mathrm e^t)^N.
$$
Thus, 
$$
I(c)=\max\{I(t,c)\,;\,t\geqslant0\},\qquad I(t,c)=tc-\log(1-p+p\mathrm e^t).
$$
Note that the condition $c\gt p$ ensures that the derivative of $I(\cdot,c)$ at $0$ is positive, thus $I(c)\gt0$ (of course, one can also get an explicit formula for $I(c)$ and check this on the formula).
