I cannot work out the proof in one of the steps. The following is copied from the original paper “Balls into Bins” — A Simple and Tight Analysis:
The case when $n\log n \ll m \leq n \text{polylog}(n)$. Assume $m = gn\log n$ where $g = g(n) \leq \text{polylog}(n)$ tends to infinity arbitrarily slowly. Then $k_{\alpha} = g \log n(1+ \alpha \sqrt \frac{2}{g})$.
I dont know how it can get from the second line to the last (third) line in calculating the expectation:
$\log E[X] = \log n + k_\alpha (\log g + \log^{(2)} n -\log k_\alpha + 1) - g \log n + O(\log^{(2)} n)$
$ = g \log n(\frac{1}{g} + (1+\alpha\sqrt\frac{2}{g})(1-\alpha\sqrt\frac{2}{g} + \frac{\alpha^2}{g} + o(\frac{1}{g}))-1+ o(\frac{1}{g}))$
$ = \log n(1- \alpha^2 +o(1))$ <---how can we get to here from the previous step?
how do some of the terms like $\alpha^3\sqrt\frac{2}{g}$ vanish?
Thanks a lot!