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I want to show this sum converges to $ 0 $ , when $n $ tends to infinity :

$\frac { 1 } { 2 ^{n^2}}\sum _{m> \frac {n^2} {2} + cn\sqrt { n}} { {n ^2}\choose{m}}$

Is this claim true?

How to show it?

Thanks.

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    own thoughts?... – 2017-01-03
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    You may appeal to the Chebyshev inequality. – 2017-01-03
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    @SangchulLee oh right!if I define $Y $ as sum of $n^2$ 0-1 Bernoulli variables ,I get $P (|Y-n^2/2|> c n \sqrt n ) < P (Y-n^2/2|> tn/2)<1/t^2$,so that sum converges to zero. Thanks! – 2017-01-03

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