Find $\sum\limits_{k=1}^n \frac{1}{k}\binom{n}{k}$

Question

I would like to find $\sum\limits_{k=1}^n \frac{1}{k}\binom{n}{k}$. I tried to write this as an integral but I can't find the result.

Answer 1

Let $$f(x)=\sum_{k=1}^n\frac1k{n\choose k}x^k$$ We are interested in $f(1)$ and note that $$ f'(x)=\sum_{k=1}^n{n\choose k}x^{k-1}=\frac1x\sum_{k=1}^n{n\choose k}x^{k}=\frac{(1+x)^n-1}{x}$$ so that $$f(1)=f(0)+\int_0^1f'(x)\,\mathrm dx=\int_0^1 \frac{(1+x)^n-1}{x}\,\mathrm dx$$