So when I was asked a question about summations, I was kind of stumped. Now that I’ve started reading a algorithms text book, I’m running into summations all over again. In my search to re-learn them, I’ve found the following sites useful:

• Paul’s Online Math Notes – Prof. Paul Dawkins, of Lamar University, has a great math site, and has a good, short, re-introduction to summations here.
• PlanetMath’s Summation Page – PlanetMath has a good page on summations that goes a little more into detail on some summation properties and useful formulas
• Summation Formulas – Good outline of summation formulas

What I haven’t been able to find is a page explaining how the solutions for common summations is found, such as the one for a simple summation:

I’m not sure how that formula is reached, but I’m sure I’ll be trying to figure it out soon.

Update: Finally found a good description of the steps in solving the summation above, here’s my attempt at explaining it:

The simple summation above can be thought of as the series:

It can also be thought of (in reverse) as:

Now, imagine writing each of the terms of those two series above eachother, then adding the ones that are above and below eachother.  You would end up with something like:

This can be simplified by doing the math inside the parens to:

Which is simplified even further to:

At this point, you’ve basically described adding the summation twice, or

Divide each side by two, and you’ve got the solution we were looking for:

I finally found this explanation at the Math Refresher weblog.  The article I linked to goes into a lot more detail, so I would suggest reading it if you want to know more.  I tried, but my eyes are getting tired from math at ths point in time, and his equation representations are a little hard to read.