Summations / Sigma Notation
A few years ago I had a job interview with a company that ended up being pretty math focussed. I had been out of college for a while at the time, and on a day to day basis in doing software development I hadn’t had the opportunity to exercise a lot of the math that I had learned while in college. At some point in my life I had taken: Calc, Calc II, Linear Algebra, Discrete Math, and some others…but because I haven’t been using it, my math knowledge had kind of gone down the tubes.
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.
This entry was posted on Friday, May 1st, 2009 at 1:10 pm and is filed under Learning, Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
