I am reading a book on financial theory and it has been a while since I took a notation heavy math course so could someone help me out here...
Now with this equation how does the sum work? Do I do the factorials first? Is it a sum for each value from 0 to i? Is everything to the right of the sum sign included each iteration of the sum?
thanks!
In this particular case, yes, everything to the right of the summation sign is included in each iteration of the sum.
It is a sum for each value of i from 0 to N.
If you wanted to evaluate this at some value, you would do the summation first, and then begin to expand the resulting expression.
For example, consider the simpler expression
$$(1 + n^2)\sum\limits_{i = 1}^n i + 2.$$
evaluated at $n = 4$, say.
The first step to take would be to substitute in $4$ for $n$, giving
$$(1 + 4^2)\sum\limits_{i = 1}^4 i + 2.$$
Then, expand the summation, giving
$$\begin{align} (1 + 4^2)((1 + 2) + (2 + 2) + (3 + 2) + (4 + 2)) &= (1 + 16)(3 + 4 + 5 + 6)\\ &= (17)(18)\\ &= 306. \end{align}$$
Essentially, the summation is expanded first, then the interior of the summation is evaluated, and then that result gets combined with the outside.
You can think of a summation as implying parentheses, with the left paren just to the left of the $\sum$, and the right paren at the far end of the expression being summed over.