I want to prove for any positive integer $n$, the following equation holds: $$\sum_{k=0}^n\frac{1}{n\choose k}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}{k}$$
I tried to expand $2^k$ as $\sum_{i=0}^k{k\choose i}$ and interchange summation, also tried let $f(x)=\sum_{k=1}^{n+1}\frac{x^k}{k}$ and compute $f'(x)$. But it seems I'm not on the right path.
Note that successive terms in the sum on the left turn out have a simple expression for their sum: when $k \ne 0$, we have $$\frac{1}{\binom{n}{k-1}} + \frac1{\binom{n}{k}} = \frac{\binom{n}{k}+\binom{n}{k-1}}{\binom{n}{k-1}\binom{n}{k}} = \frac{\binom{n+1}{k}}{\binom{n}{k-1}\binom{n}{k}} = \frac{\frac{n+1}{k}\binom{n}{k-1}}{\binom{n}{k-1}\binom{n}{k}} = \frac{\frac{n+1}{k}}{\binom{n}{k}}$$
where we've used Pascal's rule and the "absorption" identity that $\binom{n}{k} = \frac{n}{k} \binom{n-1}{k-1}$ for $k \neq 0$. Applying absorption again further gives the above expression to be equal to $$\frac{\frac{n+1}{k}}{\binom{n}{k}} = \frac{\frac{n+1}{k}}{\frac{n}{k}\binom{n-1}{k-1}} = \frac{n+1}{n} \frac{1}{\binom{n-1}{k-1}}$$
This gives a strategy for evaluating the sum on the left:
$$\begin{align} 2\sum_{k=0}^n \frac{1}{\binom{n}{k}} &= \sum_{k=1}^{n} \left( \frac{1}{\binom{n}{k-1}} + \frac1{\binom{n}{k}}\right) + 2 \\ &= 2 + \frac{n+1}{n} \sum_{k=1}^{n} \frac{1}{\binom{n-1}{k-1}} \end{align}$$ or, calling the left-hand-side sum as $L_n = \sum_{k=0}^n \frac{1}{\binom{n}{k}}$, we have $$2 L_n = 2 + \frac{n+1}{n} L_{n-1}$$ $$L_n = \frac{n+1}{2n}L_{n-1} + 1$$
Calling the right-hand-side term $R_n$, we have $$\begin{align} \frac{2^{n+1}}{n+1} R_n &= \sum_{k=1}^{n+1}\frac{2^k}{k} \\ &= \frac{2^n}{n} R_{n-1} + \frac{2^{n+1}}{n+1} \end{align}$$ thus $$R_n = \frac{n+1}{2n}R_{n-1} + 1$$
and both the LHS $L_n$ and RHS $R_n$ satisfy the same recurrence and have the same initial values (check for $n=1$, say), so they are equal.
