I am having trouble with estimation of the following$\frac{n^a}{2^{n-\frac{\sqrt n+1}{2}}(1-\frac{1}{2 \sqrt n})^{n-\frac{\sqrt n-1}{2}}} $
Where $n \in N$ and $a$ is a real number greater or equal then 2.
I am having trouble with estimation of the following$\frac{n^a}{2^{n-\frac{\sqrt n+1}{2}}(1-\frac{1}{2 \sqrt n})^{n-\frac{\sqrt n-1}{2}}} $
Where $n \in N$ and $a$ is a real number greater or equal then 2.
For every $a$, this is $2^{-n+o(n)}$, hence the limit is zero.
To show this, prove that the logarithm of $n^a$, the logarithm of the remaining power of $2$ and the logarithm of the funny looking power of $1-\frac1{2\sqrt{n}}$ in the denominator, are all $o(n)$.