Paul's Online Math Notes
Calculus I (Notes) / Extras / Summation Notation   [Notes]

On August 21 I am planning to perform a major update to the site. I can't give a specific time in which the update will happen other than probably sometime between 6:30 a.m. and 8:00 a.m. (Central Time, USA). There is a very small chance that a prior commitment will interfere with this and if so the update will be rescheduled for a later date.

I have spent the better part of the last year or so rebuilding the site from the ground up and the result should (hopefully) lead to quicker load times for the pages and for a better experience on mobile platforms. For the most part the update should be seamless for you with a couple of potential exceptions. I have tried to set things up so that there should be next to no down time on the site. However, if you are the site right as the update happens there is a small possibility that you will get a "server not found" type of error for a few seconds before the new site starts being served. In addition, the first couple of pages will take some time to load as the site comes online. Page load time should decrease significantly once things get up and running however.

August 7, 2018

Calculus I - Notes
Applications of Integrals Previous Chapter  
Types of Infinity Previous Section   Next Section Constant of Integration

 Summation Notation

In this section we need to do a brief review of summation notation or sigma notation.  We’ll start out with two integers, n and m, with  and a list of numbers denoted as follows,



We want to add them up, in other words we want,




For large lists this can be a fairly cumbersome notation so we introduce summation notation to denote these kinds of sums.  The case above is denoted as follows.




The i is called the index of summation.  This notation tells us to add all the ai’s up for all integers starting at n and ending at m.


For instance,





Here are a couple of formulas for summation notation. 

1.       where c is any number.  So, we can factor constants out of a summation.

2.        So we can break up a summation across a sum or difference.


Note that we started the series at  to denote the fact that they can start at any value of i  that we need them to.  Also note that while we can break up sums and differences as we did in 2 above we can’t do the same thing for products and quotients.  In other words,






Here are a couple of nice formulas that we will find useful in a couple of sections.  Note that these formulas are only true if starting at .  You can, of course, derive other formulas from these for different starting points if you need to.







Here is a quick example on how to use these properties to quickly evaluate a sum that would not be easy to do by hand.


Example 1  Using the formulas and properties from above determine the value of the following summation.



The first thing that we need to do is square out the stuff being summed and then break up the summation using the properties as follows,



Now, using the formulas, this is easy to compute,



Doing this by hand would definitely taken some time and there’s a good chance that we might have made a minor mistake somewhere along the line.

Types of Infinity Previous Section   Next Section Constant of Integration
Applications of Integrals Previous Chapter  

Calculus I (Notes) / Extras / Summation Notation    [Notes]

© 2003 - 2018 Paul Dawkins