Let $x_i$ be positive number satisfying $\sum x_i = 1$, what is $\sum ix_i$ Let $x_i$ be a positive number for each $i \in \{1, 2, 3 \dots \}$  such that $\sum_{i=1}^\infty x_i = 1$ is there a closed formula for 
$$\sum_{i = 1}^\infty ix_i$$?

There isn't a formula independent of the $x_i$. The series does not always converge (let $x_i=\frac{6}{\pi^{2}i^{2}}$), but can converge (let $x_{i}=\frac{1}{2^{i}}$). I would guess the only limitation on the value of the series is that it is greater than 1.
