A number-theoretical estimation-inequality I have some trouble understanding the following number-theoretical estimation:

$$\sum_{k\le \sqrt{n}} (1-k^2/n)^{1+o_n(1)}=n^{1/2+o(1)} \
 (n\to\infty),$$ where $o_n(1)$ denotes a $o(1)$ function depending
  only on $n$.

The side $\ge$ is obvious, but the other one gives me trouble.
Indeed, without the $f(k)=o_n(1)$ term, I'd try to apply integral comparison or such, but having this unknown function I can't do that.
My idea was to make it disappear in an inequality, but since I don't known anything about the sign of $f$ I'm blocked. For example, if $f\ge 0$, I could argue that
$$(1-k^2/n)^{f(k)}\le 1,$$
but it is not necessarily true.

Divide both sides by $n^{1/2}$:
$$
\lim_{n\to\infty}\sum_{k\le\sqrt{n}}(1-k^2/n)^{1+o(1)}\frac1{\sqrt{n}}=\int_0^1(1-x^2)\,\mathrm{d}x=\frac23
$$
since the left hand side is a Riemann sum with partition size $\frac1{\sqrt{n}}$. The exponent tends to $1$ as $n\to\infty$.
Therefore,
$$
\sum_{k\le\sqrt{n}}(1-k^2/n)^{1+o(1)}\frac1{\sqrt{n}}\sim\frac23n^{1/2}=n^{1/2+\frac{\log(2/3)}{\log(n)}}=n^{1/2+o(1)}
$$
