Squared Hellinger distance between Binomial(n,p) and Binomial(n+1,p)
Write $$\begin{align}
h^2(B(n,p), B(n+1,p))&amp; = 2 \left(1 - \sum_{i=0}^{n+1}\sqrt{{n \choose i}p^i(1-p)^{n-i}{n+1 \choose i} p^i (1-p)^{n+1-i}}\right)\\
&amp;=2\left(1 - \sum_{i=0}^{n+1}\sqrt{\left({n+1 \choose i}p^i(1-p)^{n+1-i}\right)^2 \frac{n+1-i}{(n+1)(1-p)}} \right)\\
&amp;=2 \left(1 - \sum_{i=0}^{n+1}{n+1 \choose i}p^i(1-p)^{n+1-i}\sqrt{ \frac{n+1-i}{(n+1)(1-p)}} \right)\\
&amp;=2\left(1-\frac{1}{\sqrt{(n+1)(1-p)}} \mathbb{E}(\sqrt{Z})\right),\\
\end{align}$$
where $Z \sim B(n+1, 1-p)$. By Taylor approximation (see this), we have
$$\mathbb{E}(\sqrt{Z}) \ge  \sqrt{(n+1)(1-p)} - \frac{p}{2 \sqrt{(n+1)(1-p)}}. $$
Combining with the above, we get
$$ h^2(B(n,p), B(n+1,p)) \le 2 \left(1 - \left(1 - \frac{p}{2 (n+1)(1-p)}\right)\right) = \frac{p}{(n+1)(1-p)}.$$