I have the following series:
S3 = 1/1 + 1/2 + 1/3 + 1/1*2 + 1/1*3 + 1/2*3 + 1/1*2*3
The question is to find a formula to produce this series (Sn)
I'm kinda stuck here. I know you can determine the number of subsets of a set A by 2^|A| but I'll need to do something with faculties or something.
Could anyone help me out please?
(Ps: this is not homework, it's an exercise I'm trying to make before my final next week)
As $$S_3=\frac11+\frac12+\frac13+\frac1{1\cdot2}+\frac1{2\cdot3}+\frac1{3\cdot1}+\frac1{1\cdot2\cdot3}$$ $$=\left(1+\frac11\right)\left(1+\frac12\right)\left(1+\frac13\right)-1$$
$$S_n=\prod_{1\le r\le n}\left(1+\frac1r\right)-1=\prod_{1\le r\le n}\left(\frac{r+1}r\right)-1$$ $$=\frac{2\cdot3\cdots n\cdot(n+1)}{1\cdot2\cdots(n-1)\cdot n}-1=\frac{n+1}1-1=n$$ for integer $n\ge1$
More generally, if $$S_3=\sum f(r_i)+\sum f(r_i)f(r_j)+\sum f(r_i)f(r_j)f(r_k)=-1+\prod_{1\le i\le 3}\{1+f(r_i)\}$$
$$S_n=-1+\prod_{1\le i\le n}\{1+f(r_i)\}$$
