Equivalent summations

Question

If the function $p(r)$ maps from the positive integers to the non-negative real numbers and has the property that $\sum_{r=1}^\infty p(r) = 1$, and $x_1, x_2, ... x_n$ is a sequence for which $X = \sum_{r=1}^\infty x_r p(r)$ is well-defined and the summation $\sum_{r=1}^\infty x_r^2p(r)$ is well defined, which of the following equals $\sum_{r=1}^\infty (x_r - X)^2p(r)$?

1)$[\sum_{r=1}^\infty (x_r^2 p(r)]-X^2$

2) $\sum_{r=1}^\infty (x_r^2 + X^2)p(r)$

3) $\sum_{r=1}^\infty (x_r^2 +2x_r X - X^2)p(r)$

4) $\sum_{r=1}^\infty (x_r - X)^2p(r)$ may not be well defined

Answer 1

The browser is not rendering your webpage properly and your tags are visible.

 

It looks like you want the sum of the 1st N integers, which is

 

 N(N+1)(2N+1)/6

 

Please repost