From their description:

Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

I’ve done a few of the first problems, implemented them all in ruby so far (or just did the math required), and they’ve definitely been interesting and fun to work on.  4 down, 242 to go.  All of the problems are math related, and you are basically presented with the problem and a field to input your answer.  After you supply the correct answer you’re given a link to the forum (most of which are locked now) where people have discussed their approaches to solving the problem.

If you’re looking to brush up on some math skills, or just need some interesting problems to throw some code at, this site provides some great things to work on.

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.