I was recently reading a text book on random walk. In the proof of a local central limit theorem the book used the following step: $e^{-2k^2/(n+k)} = e^{-2k^2/n} \space exp\{\frac{2k^3}{n^2} +O(\frac{k^4}{n^3})\}$. And similarly, there is: $(1-\frac{k^2}{n^2})^{-1/2} = exp\{O(\frac{1}{n}+\frac{k^4}{n^3})\}$, and the latter case is deduced using the fact that $k^2/n^2 \leq max\{(1/n),(k^4/n^3)\}$. I think the expanding of complex logarithm with power series is used in both cases, but I'm still quite confused how these equalities are deduced.
Could anyone help me with it?
The big O notation above is defined as follows:"write $f(n,x) = g(n,x) +O(h(n))$ to mean that there exists a constant c such that for all n $|f(n,x)-g(n,x)| \leq c|h(n)|$, where c denotes a positive constant depending on the increment distribution"
Consider $$a_n=e^{-\frac{2 k^2}{k+n}}\implies \log(a_n)=-\frac{2 k^2}{k+n}$$ Now, if $n \gg k$, use Taylor expansion $$\log(a_n)=-\frac{2 k^2}{n}+\frac{2 k^3}{n^2}+O\left(\frac{1}{n^3}\right)$$ Exponentiate to get $$a_n=e^{-\frac{2 k^2}{n}}\exp\left(\frac{2 k^3}{n^2}+O\left(\frac{1}{n^3}\right) \right)$$