Question about closed-form of a series

Question

my question is about the below series: $$\sum_{i=1}^n (1+x_i)^{-1}$$ and $x_i$ are n variables. Does it have a closed-form? or a simple form?

I would be appreciated if anyone can help me.

Answer 1

Let $p(x)= (x + x_1)(x+x_2)\cdots (x+x_n).$ Then by taking logarithms and differentiating we have

$$ \frac{p'(x)}{p(x)} = \sum_{i=1}^n \frac{1}{x+x_i}$$

So your sum is $p'(1)/p(1)$.