2

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

$$\sum_{i = 1}^\infty ix_i$$?

user99888
  • 21
  • No, it can take almost any value. p.e. if $x_{k}=1$ and $x_{i}=0$ for the other $i$ then it equals $k$. If the $x_{i}$ are nonnegative then you can recognize the expectation of a discretely distributed and positive random variable $X$ in it. – drhab Oct 09 '13 at 20:13

1 Answers1

0

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.

Albert Zhang
  • 606