Let's start by looking at this graph
DiscretePlot[
Sum[Binomial[n, i]/i^((n + 1)/2), {i, 1, n}]/n, {n, 500, 550}]

So we have the hypothesis that the sum increases like $n$. Let's use Mathematica to prove that the sum $S$ divided by $n$ goes to 1.
We have
Limit[1/n Binomial[n, i]/i^n, n -> Infinity, Assumptions -> i >= 2]
0
So all but the first term go to 0. But we have a lot of such terms, so this is not enough to say the "rest of the sum" $R$ goes to 0. We prove that the sum from i=3
onwards goes to 0.
We have
Limit[n Binomial[n, n/2]/3^n, n -> Infinity]
0
This proves that the sum from i=3 onwards goes to 0, as the Binomial coefficient is maximal when i=n/2
, 1/(i^n)
is maximal when i=3
and we have n-2
~ n
terms.
The second term also goes to 0, so indeed the "rest of the sum", i.e. $R$, goes to 0.
The first term is
n == Binomial[n,1]*1^x
So we have that
$$\lim_{n\rightarrow \infty} \frac{S}{n} = \lim_{n\rightarrow \infty} \frac{n}{n} + \lim_{n\rightarrow \infty} \frac{R}{n} = 1+0 = 1$$
f[n_] := Sum[Binomial[n, i]/i^((n + 1)/2), {i, 1, n}]; a = Table[f[n], {n, 1, 100}]
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