Summation Notation: Rules & Examples

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  • 0:01 Definition
  • 1:24 How to Use It
  • 3:04 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Did you know that the summation notation is also called the sigma notation? You will learn about this unique and simple notation in this lesson, along with how it helps you in writing shorter equations.


The summation notation is a way to quickly write the sum of a series of functions. It is also called sigma notation because the symbol used is the letter sigma of the Greek alphabet. Specifically, sigma is the Greek equivalent to the capital letter S. Why do you think we use this particular letter of the Greek alphabet for the summation notation? You guessed it, because the words sum and summation begin with the letter S.

summation notation

The symbol itself is very unique and simple. And we will be adding additional notes to the symbol to tell us more about what kind of summation we need. When everything is written down, the full summation notation will have little numbers above and below the sigma symbol and a function to the right of the symbol.

The full summation notation.
summation notation symbol

In its full glory, this notation tells us that we are going to add the function f(i) evaluated at 1, then 2, all the way up to the number n.

This tells us to evaluate the function at 1, 2, all the way up to n. Then we are to add it all up.
summation notation full

Can you see where the sum comes into place? Yes, the notation tells us to evaluate the function at each number between the lower and upper number, and then we are to add or sum up our results together to get the answer. The little numbers on top and below the sigma symbol are called your index numbers. They tell you at what number to start evaluating and at what number to stop evaluating.

How to Use It

Once you understand the notation, using this symbol becomes a piece of cake. Let's start with a simple case just to show you how it all comes together. Let's summate the function f(i)=i beginning with the number 1 and ending with the number 4.

This is how we summate the function from 1 to 4.
summation notation example

Now, that was easy. Can you tell me what just happened? In this simple example, our function happens to be f(i)=i. Evaluating this function at each number 1, 2, 3, and 4 gives us 1, 2, 3, and 4 respectively, since the function tells us to simply plug in our index numbers wherever we see the letter i. The summation notation tells us we are to sum it all up, which we have done, to get our answer 10.

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