<s>[INST] How prove this inequality $\sum_{k=1}^{n}\frac{2k-1}{k\binom{n}{k}}\ge \frac{n}{2^{n-1}}$ 
let $1\le k\le n,k,n\in N^{+}$, show that
  $$\sum_{k=1}^{n}\dfrac{2k-1}{k\binom{n}{k}}\ge \dfrac{n}{2^{n-1}}$$

I know this $$\sum_{k=1}^{n}(2k-1)=n^2$$
and $$\sum_{k=1}^{n}k\binom{n}{k}=n\cdot 2^{n-1}$$
I want Use Cauchy-Schwarz inequality .
$$\left(\sum_{k=1}^{n}\dfrac{2k-1}{k\binom{n}{k}}\right)\left(\sum_{k=1}^{n}k\binom{n}{k}\right)\ge (\sum_{k=1}^{n}\sqrt{2k-1})^2$$
then
$$\sum_{k=1}^{n}\dfrac{2k-1}{k\binom{n}{k}}\ge\dfrac{(\sum_{k=1}^{n}\sqrt{2k-1})^2}{n\cdot 2^{n-1}}$$
Now we must prove
$$\sum_{k=1}^{n}\sqrt{2k-1}\ge n?$$
maybe can use integral inequality to prove it.
I can't prove this.Thank you
 [/INST] Another starting point is the inequality $\boxed{\binom{n}{k} \leq 2^n}$  This trivial starting point allows us to deduce that
$$  \sum_{k=1}^n \binom{n}{k}^{-1} \geq \frac{n}{2^n} $$
If we plugged this into the original inequality we fall short of what we're trying to prove:
$$ \sum_{k=1}^n \left( 1 - \frac{1}{2k}\right)\binom{n}{k}^{-1} \geq
2^{-n} \sum_{k=1}^n \left( 1 - \frac{1}{2k}\right) \geq \mathbf{\color{blue}{\frac{n - \tfrac{1}{2}\log n}{2^n}}}
$$
Or we can try it the other way, but we still fall a little bit short.
$$ \sum_{k=1}^n \left( 1 - \frac{1}{2k}\right)\binom{n}{k}^{-1} =
\sum_{k=1}^n \left( k - \frac{1}{2}\right)\frac{1}{n}\binom{n-1}{k-1}^{-1} 
= \frac{1}{n 2^{n-1}}\sum_{k=1}^n \left( k - \frac{1}{2}\right)
= \mathbf{\color{blue}{\frac{n- \frac{1}{n}}{ 2^n} }}
$$

I hope improve $\frac{1}{2^n}\binom{n}{k} \leq 1$, perhaps by a constant that depends on $k$.

In fact, we can use the Arithmetic mean - Harmonic mean inequality - or possibly Jensen's inequality - to get:
$$ \sum_{k=1}^n \left( 1 - \frac{1}{2k}\right)\binom{n}{k}^{-1} \geq 
\frac{n^2}{\sum_{k=1}^n \left( 1 - \frac{1}{2k}\right)^{-1}\binom{n}{k}}
\geq 
\frac{n^2}{2 \sum_{k=1}^n \binom{n}{k}}
> \mathbf{\frac{n^2}{2^{n+1}} }$$
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