TITLE: Can a sum of products be split as a product of two sums?
QUESTION [18 upvotes]: I have 
$$\sum_k^n P_k x_k$$   
Am I allowed to split it up into two sums so I have it like 
$$\sum_k^n P_k  \sum_k^nx_k$$

REPLY [13 votes]: Assuming P depends on k and not n then the summation
$$
\sum_k P_k x_k
$$ 
Creates terms like
$$
P_1 x_1 + P_2 x_2 + \ldots + P_n x_n
$$
But, if you took $P_k$ out then you would have two independent sums:
$$
\sum_k P_k \sum_k x_k
$$
Producing two separate sums multiplying each other like
$$
(P_1 + P_2 + \ldots + P_n) (x_1 + x_2 + \ldots + x_n)
$$
And this obviously produces terms like $P_1 x_1, P_1 x_2,\ldots$ or as produced by the double sum written like:
$$
\sum_i \sum_j P_i x_j
$$
So the original summation is definitely not the same as this double sum.