Exercise C.5.3

$\star$ Show that

$$ \sum_{i=0}^{k-1}\binom{n}{i}a^i < (a+1)^n \frac{k}{na - k(a+1)}b(k;n,a/(a+1)) $$

for all $a > 0$ and all $k$ such that $0 < k < na/(a+1)$.

$$ \begin{aligned} (a+1)^n \frac{k}{na - k(a+1)}b(k;n,a/(a+1)) &= (a+1)^n \frac{k \frac{1}{a+1}}{\frac{na-ak-a}{a+1}} b(k;n,a/(a+1)) & \text{(C.4)} \\ &> (a+1)^n \sum_{i=0}^{k-1}b(i;n,a/(a+1)) \\ &= (a+1)^n \sum_{i=0}^{k-1}\binom{n}{i}\frac{a^i}{(a+1)^i}\frac{1}{(a+i)^{n-i}} \\ &= (a+1)^n \frac{1}{(a+1)^n} \sum_{i=0}^{k-1}\binom{n}{i}a^i \\ &= \sum_{i=0}^{k-1}\binom{n}{i}a^i \end{aligned} $$