Exercise C.5.4

$\star$ Prove that if $0 < k < np$, where $0 < p < 1$ and $q = 1 - p$, then

$$ \sum_{i=0}^{k-1}p^iq^{n-i} < \frac{kq}{np-k} \bigg(\frac{np}{k}\bigg)^k \bigg(\frac{nq}{n-k}\bigg)^{n-k} $$

$$ \begin{align} \sum_{i=0}^{k-1}p^iq^{n-i} &= \Pr\{X < k\} & \text{(C.4)} \\ &< \frac{kq}{np - k} b(k;n,p) & \text{(C.1)} \\ &< \frac{kq}{np - k} \bigg(\frac{np}{k}\bigg)^k \bigg(\frac{nq}{n-k}\bigg)^{n-k} \end{align} $$