We introduce sigma notation.

The notation itself

Sigma notation is a way of writing a sum of many terms, in a concise form. A sum in sigma notation looks something like this:

The (sigma) indicates that a sum is being taken. The variable is called the index of the sum. The numbers at the top and bottom of the are called the upper and lower limits of the summation. In this case, the upper limit is , and the lower limit is . The notation means that we will take every integer value of between and (so , , , , and ) and plug them each into the summand formula (here that formula is ). Then those are all added together.

Try one on your own.

Write out what is meant by the following (no need to simplify):

Let’s try going the other way around.

Try one on your own.

Write the following sum in sigma notation. We write:

Calculating with sigma notation

We want to use sigma notation to simplify our calculations. To do that, we will need to know some basic sums. First, let’s talk about the sum of a constant. (Notice here, that our upper limit of summation is . is not the index variable, here, but the highest value that the index variable will take.) This is a sum of terms, each of them having a value . That is, we are adding copies of . This sum is just . The other basic sums that we need are much more complicated to derive. Rather than explaining where they come from, we’ll just give you a list of the final formulas, that you can use.

Formula 1. .

Formula 2. .

Formula 3. .

Now that we have this list, let’s use them to compute.

Because sigma notation is just a new way of writing addition, the usual properties of addition still apply, but a couple of the important ones look a little different.

Commutativity and Associativity:
Distribution:

Try one on your own.

Find the value of the sum