Let $X$ be an uniform random variable between $[0,K]$. I want to find an upper bound for a state that $X> \alpha$. So I have used blow Chernoff bound:
\begin{align*} \mathbb{P}(X\ge \alpha)&\le \frac{\mathbb{E}[e^{tX}]}{e^{\alpha t}} \end{align*}
And after calculation, I have found the blow equation :
\begin{align*} \mathbb{P}(X\ge \alpha)&\le \frac{e^{t}(\frac{1-e^{tk}}{1-e^t})}{k.e^{\alpha t}} \end{align*}
But, what is the simply form of this equation ? is it possible to write or approximate that based on a $e$ like $e^\beta$ which $\beta$ is based on $\alpha$ and $k$?