Upper bound for $\sum_{i = 0}^{k-1} {n \choose i} (1 - \varepsilon)^i\varepsilon^{n-i}$ What is tight upper and lower bound for following expression where $0 < \varepsilon < 1$ and $1\leq k \leq n$.
$\sum_{i = 0}^{k-1} {n \choose i} (1 - \varepsilon)^i\varepsilon^{n-i}$

For a simpler solution, use the appximation to $\binom{n}{k} = \frac{n^{k}}{k!}$, then the summand becomes
$$
(1-\varepsilon)^n \frac{n^{i}\varepsilon^{i}(1-\varepsilon)^{-i}}{i!}
$$
then the upper bound can be easily found by using partial Taylor series expansion for the exponential function:
$$
S_{n} \leq(1-\varepsilon)^{n}\exp\bigg(\frac{n \varepsilon}{1-\varepsilon}\bigg)
$$
