Summation, or Sigma, Notation

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

There are many situations in which we need to sum or add a large number of numbers. Because this happens so often, mathematicians have adopted some notation -- called summation notation or sometimes sigma notation (because the Greek letter sigma is used) for use in this situation. Computer algebra systems like Maple, Mathematica, and the TI-92 have similar notation.

We illustrate this notation with an example. Suppose that we want to find the sum

1 + 4 + 9 + ... + 10,000

Notice that we are adding 100 terms. We will use the notation a_j for the j-th term in this sum. Notice that the terms can be described by the formula

a_j = j²

Using summation notation this sum is written

Notice that this notation has several elements labeled in the figure below.

An index variable: In this case the index variable is j. This variable is used to number or label each term.
A number indicating the start of the sum -- that is, the number of the first term. In this case the start is 1. The start is usually either zero or one but it can be anything. For example, we often start a summation with the current year.
A formula describing each term. In this case the j-th term is j².
A number indicating the end of the sum -- that is, the number of the last term. In this case the end is 100.

The upper case Greek letter sigma which is the Greek equivalent of the English letter ess is used to denote summation.

Look at your CAS window to see how summation is denoted for your CAS system.

Check Your Understanding

Write the sum
```
1 + 2 + 3 + ... + 10
```
using sigma notation and then compute it using your computer algebra system. answer
Write the sum
```
1 + 2 + 3 + ... + 100
```
using sigma notation and then compute it using your computer algebra system. answer
Write the sum
```
1 + 3 + 5 + ... + 99
```
using sigma notation and then compute it using your computer algebra system. answer
Write the sum
```
1 + 2 + 4 + 16 + ... + 512
```
using sigma notation and then compute it using your computer algebra system. answer
Write the sum
```
1 + (1/2) + (1/4) + ... + (1/1024)
```
using sigma notation and then compute it using your computer algebra system. answer