Asymptotics of a real sequence

Question

Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence with $a_n\in O(n^d)$ $(d\in (-1,0))$. Now we consider the expression $$ b_n:=(1-\sqrt{1-a_n}).$$ Is $b_n\in O(\sqrt{n^d})$?

Thanks!

Answer 1

Yes. Even better, it is in $O(n^d)$. We have

$$\sqrt{1-x} = 1 - \frac{x}{2} - \frac{x^2}{8} + O(x^3)$$

for $\lvert x\rvert < 1$ by the Taylor expansion, so

$$1 - \sqrt{1-a_n} = 1 - (1 - \frac{a_n}{2} + O(a_n^2)) = \frac{a_n}{2} + O(a_n^2).$$

Since $d < 0$, $n^d$ is dominated by $\sqrt{n^d}$.