I have to understand what is this expression $\sum_{A\subset[n]}\prod_{i\in A}1/i$ where $[n]=\{1,\ldots,n\}$. And then prove it. I was using a very complicated method to understand what this expression is.
The hint of the book is: express the sum as a product.
My method is: $a_n:=\sum_{A\subset[n]}\prod_{i\in A}1/i$ so we have $a_{n+1}=(1/(n+1)+1)a_n$, so if we call $y(x)=\sum_{i=1}^\infty a_nx^n$ we have that $y$ satisfies $y^\prime=(2y+1)/(1-x)$ but I don't know how to continue. I think there is a very very simpler way to compute this expression.
Could any of you help me, please? You can also give me the result without a proof, I will prove it by induction.