TITLE: Rules for Product and Summation Notation
QUESTION [20 upvotes]: When we deal with summation notation, there are some useful computational shortcuts, e.g.:
$$\sum\limits_{i=1}^{n} (2 + 3i) = \sum\limits_{i=1}^{n} 2 + \sum\limits_{i=1}^{n} 3i = 2n + \sum\limits_{i=1}^{n}3i$$
However, I don't think I know all the useful shortcuts here. Are there other computational tricks one should be aware of? What's a good way for thinking about this?
More importantly, consider product notation:
$$\prod\limits_{i=1}^{n} (\sqrt{2} - \sqrt[n]{2})$$
I don't know what the shortcuts here are. What are some of the more effective ways of attacking such computations?

REPLY [5 votes]: Summation of product of two functions.
$$ \sum_{i=1}^{n} f(x) g(x) $$ = $$ \sum_{i=1}^{n}f(x)∑g(x) - 
 [ f(x)∑g(x-1) + f(x-1)∑g(x-2) + f(x-2)∑g(x-3) + .... + f(2)∑g(1) + g(x)∑f(x-1) + g(x-1)∑f(x-2) + g(x-2)∑f(x-3) + .... + g(2)∑f(1)]$$
This is not the simplest form yet. Someone please help me to simplify and retype.