Here, $x$ is a fixed number, and $n$ goes to infinity.
I want to prove that $(1+ \frac{x+\frac{1}{2}\log 4\pi - \frac{1}{2} \log \log n}{\log n} + o(\frac{1}{\log n}))^{\frac{1}{2}} = 1+\frac{x-\frac{1}{2} \log 4\pi - \frac{1}{2} \log \log n}{2\log n} + o(\frac{1}{\log n})$.
I know that $\sqrt{1+x} = 1+\frac{x}{2} + o(x)$, but plugging in, I get
$1+\frac{x-\frac{1}{2} \log 4\pi - \frac{1}{2} \log \log n}{2 \log n} + o(\frac{1}{\log n}) + o( \frac{x-\frac{1}{2} \log 4\pi - \frac{1}{2} \log \log n}{2\log n} + o(\frac{1}{\log n}))) = 1+\frac{x-\frac{1}{2} \log 4\pi - \frac{1}{2} \log \log n}{2\log n} + o(\frac{1}{\log n}) + o(\frac{\log \log n}{\log n}) = 1+\frac{x-\frac{1}{2} \log 4\pi - \frac{1}{2} \log \log n}{2\log n} + o(\frac{\log \log n}{\log n})$
What did I do wrong? This is in little-o notation.
Thank you.
Here is the relevant section:
