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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.

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    Paula: Did I get the missing right parenthesis in the right place? Also, in the exponenents do you want the $+1$ and $-1$ to go inside the square roots?2012-04-30
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    By _estimation_, do you mean the limit of the expression as $n\to\infty$?2012-04-30
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    @: Brian Scott, thank you, your corrections are right.2012-04-30

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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 $o(n)$.