I have to evaluate an infinite product in the form
$\prod_{i=0}^{\infty} \left( 1 + f_i \right) $
where the $f_i$ are real numbers. I came out with the following trick: by noting that, calculating all the products, I will get all the possible 0-order terms (the only term of that kind being $1$), all possible 1st-order terms ($f_1 + \cdots + f_n$), all possible 2nd-order terms ($f_1 \cdot f_2 + f_1 \cdot f_3 \cdots $) and so on... and by noting that each term shows up exactly one time, and that permutations won't show up in the product, I wrote: $ \prod_{i=0}^{\infty} \left( 1 + f_i \right) = \left(1 + \sum_i^\infty f_i + \frac{1}{2!} \sum_i^\infty f_i \sum_j^\infty f_j + \cdots \right) = \exp \left( \sum_i^\infty f_i\right) $
the combinatorial factors $\frac{1}{n!}$ arise from the no-permutations condition... However Wikipedia says that this is not the correct result, but only an upper bound... I can't see why it is so. (first question)
(Second question) I would like to demonstrate that, given the right assumptions, what I found is the right result. The extra assumption I can make on $f_i$ is that they are positive real numbers; also I know for sure that the result will converge. Is it enough?
Thanks in advance!