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I am trying to evaluate the limit $\lim_{n\to\infty} \frac{n}{2^n}\sum_{k=1}^n\frac{1}{k}\binom{n}{k}$.

While numeric simulations show that the limit gets close to 2 when n goes to $\infty$, it is not clear to me how to simplify the sum $\sum_{k=1}^n\frac{1}{k}\binom{n}{k}$. Is there any known expression for the result of this summation?

Gary
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user13124
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    Hint: we wish to show$$\int_0^1\sum_{k=1}^n\binom{n}{k}x^{k-1}dx=\int_0^1\frac{(1+x)^n-1}{x}dx=\int_0^1\int_0^1n(1+xy)^{n-1}dxdy$$is asymptotic to $\frac{2^{n+1}}{n}$, i.e. $\int_0^1\int_0^1(1+xy)^{n-1}dxdy\sim\frac{2^{n+1}}{n^2}$. – J.G. Mar 09 '21 at 19:40

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