There are $2^{3463}-1$ non-empty subsets to... I'm clueless as to what this problem even wants me to do. Set theory is not my strength in math.
Problem: There are  $2^{3463}-1$ non-empty subsets to $\{\frac{1}{1},\frac{1}{2},\frac{1}{3},...,\frac{1}{3463}\}$. For each such subset, form the product of all elements in the subset. What is the sum of all these products?
 A: To simplify, generalize:

There are $2^n$ subsets to $S:=\{a_1,\ldots,a_n\}$. For each such subset, form the product of all elements in the subset (with the empty product equal to $1$). What is the sum of all these products?

Let $f(S)$ denote the desired result.
For $n=0$, there is only one subset, the empty subset; its product is $1$, the sum is $1$, so $f(\emptyset)=1$.
Assume we know how to handle the case of $n$ elements. What is the result for $n+1$? Subsets of $\{a_1,\ldots,a_{n+1}\}$ are either subsets of $\{a_1,\ldots,a_{n}\}$ or are subsets of $\{a_1,\ldots,a_{n}\}$ with $a_{n+1}$ added to them. The sum over the products of the first kind is $f(\{a_1,\ldots,a_n\})$. For those of the second kind, each summand gets multiplied by $a_{n+1}$; hence so does the sum. We conclude
$$f(\{a_1,\ldots,a_{n+1}\}) =(1+a_{n+1})\cdot f(\{a_1,\ldots,a_{n}\}).$$
From this we readily see (formally: by induction) that
$$f(\{a_1,\ldots, a_n\})=\prod_{k=1}^n(1+a_k).$$
Use this formula for the set of the problem statement (and subtract $1$ for the empty set not to be counted). Be ready for a surprise (i.e., first try it with $3463$ replaced by something smaller, for example $5$).
