Sum of single summation to double summation

Question

I have a equation, $f(U)=\sum_{k=1}^pb_{1k}A_{k1}U_1+\sum_{k=2}^pb_{1k}A_{k2}U_2+...+\sum_{k=p}^pb_{1k}A_{kp}U_p$

How can I convert it into double summation $\sum\sum$, how to set the limit?

Answer 1

It should be this:

$f(U)=\sum_{t=1}^p\sum_{k=t}^pb_{1k}A_{kt}U_t$

Semantically this is the same as:

$f(U)=\sum_{t=1}^p(\sum_{k=t}^pb_{1k}A_{kt}U_t)$

(assuming you missed $\ \dots\ $ in your sum)

So for $t=1$ we have: $\sum_{k=1}^pb_{1k}A_{k1}U_1$
Then for $t=2$ we have: $\sum_{k=2}^pb_{1k}A_{k2}U_2$
and so on.