Why do we have that $ \sqrt{ 1 + \frac{- \log \log n}{\log n} + o(\tfrac{1}{\log n})} = 1 + \frac{- \log \log n}{2\log n} + o(\tfrac{1}{\log n}) $? I don't understand, why the following is true:
$$
\sqrt{ 1 + \frac{-  \log \log n}{\log n} + o\left(\tfrac{1}{\log n}\right)} = 1 + \frac{-  \log \log n}{2\log n} + o\left(\tfrac{1}{\log n}\right).
$$
I know that the $1/2$-factor on the RHS comes from the series expansion of the Binomial series. But I don't see why the little $o$-factor remains the same. Shouldn't the new $o$-factor on the RHS be
$$
o\left(\tfrac{\log \log n}{\log n}\right)  + o\left(\tfrac{1}{\log n}\right)
$$
?
This problem is taken from the book "Extremes and related properties of random sequences and processes" form Leadbetter, Lindgren, Rootzén. It seems like it is an error.

 A: Note that by definition
$$o(\tfrac{\log \log n}{\log n})  + o(\tfrac{1}{\log n})=o(\tfrac{\log \log n}{\log n}) $$
indeed, with $\omega_i(n)\to 0$
$$o(\tfrac{1}{\log n})=\omega_1(n)\cdot\tfrac{1}{\log n}$$
$$o(\tfrac{\log \log n}{\log n})=\omega_2(n)\cdot\tfrac{\log \log n}{\log n}$$
then
$$o(\tfrac{\log \log n}{\log n})  + o(\tfrac{1}{\log n})=\omega_1(n)\cdot\tfrac{1}{\log n}+\omega_2(n)\cdot\tfrac{\log \log n}{\log n}=\omega_3(n)\tfrac{\log \log n}{\log n}$$
indeed
$$\omega_3(n)=\omega_1(n)\cdot\tfrac{1}{\log \log n}+\omega_2(n)\to 0$$
Note also that we can write correctly for the little-o term
$$o(\tfrac{\log \log n}{\log n})  + o(\tfrac{1}{\log n})$$
while the given little-o term 
$$o(\tfrac{1}{\log n})$$
is uncorrect indeed let
$$o(\tfrac{\log \log n}{\log n})  + o(\tfrac{1}{\log n})=\omega_1(n)\cdot\tfrac{1}{\log n}+\omega_2(n)\cdot\tfrac{\log \log n}{\log n}=\omega_4(n)\tfrac{1}{\log n}$$
then
$$\omega_4(n)=\omega_1(n)+\omega_2(n)\cdot \log \log n$$
but we can't conlcude that
$$\omega_2(n)\cdot \log \log n \to 0$$
A: The correct one is
\begin{align}
\sqrt{ 1 + \frac{-\log \log n}{\log n} + o\left(\frac{1}{\log n}\right)} &= 1 + \frac{1}{2}\left(\frac{-\log \log n}{\log n} + o\left(\frac{1}{\log n}\right)\right)+o\left(\frac{-\log \log n}{\log n} + o\left(\frac{1}{\log n}\right)\right)\\
&= 1 + \frac{1}{2}\left(\frac{-\log \log n}{\log n} + o\left(\frac{1}{\log n}\right)\right)+o\left(\frac{\log \log n}{\log n} \right)\\
&= 1 - \frac{\log \log n}{2\log n} + o\left(\frac{1}{\log n}\right)+o\left(\frac{\log \log n}{\log n} \right)\\
&= 1 - \frac{\log \log n}{2\log n} +o\left(\frac{\log \log n}{\log n} \right).\\
\end{align}
