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High School Precalculus: Homework Help Resource32 chapters | 273 lessons

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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*.

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.

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*.

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.

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.

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.

We have been using the letter *i* as part of our notation, but we can easily use any letter we want depending on our function. Let's try another example. We will have a function *f*(*n*) which we will evaluate from 1 to 4.

Notice how our index letter has changed to the letter *n*. This letter needs to match the letter of our function. In this case, both our function and our index letter is the letter *n*. This time our notation is written slightly differently than before, but that's okay. It still tells us to perform the same operations. In this case, it tells us to perform the operations on our function *f*(*n*) where our function equals *n*+1. Let's see what happens when we do that.

When we evaluate the function for each of the numbers, we need to plug the number into our function. In this example, our function is *n*+1, so plugging in each number gives us 1+1, 2+1, and so on. Summing it all up gives us 14 as our answer.

Treat the **summation notation**, which is a way to quickly write the sum of a series of functions or **sigma notation**, the alternate name of the summation notation, because the symbol used is the letter sigma of the Greek alphabet, as your friend. It really does help in simplifying your life just a little bit more. You can view it as a shorthand symbol that tells you to add everything up after evaluating your function at each of the index numbers.

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High School Precalculus: Homework Help Resource32 chapters | 273 lessons

- What is a Mathematical Sequence? 5:37
- Introduction to Sequences: Finite and Infinite 4:57
- How to Use Factorial Notation: Process and Examples 4:40
- Summation Notation and Mathematical Series 6:01
- How to Use Series and Summation Notation: Process and Examples 4:16
- Understanding Arithmetic Series in Algebra 6:17
- How to Calculate an Arithmetic Series 5:45
- Arithmetic Sequences: Definition & Finding the Common Difference 5:55
- How and Why to Use the General Term of an Arithmetic Sequence 5:01
- The Sum of the First n Terms of an Arithmetic Sequence 6:00
- How to Find and Classify an Arithmetic Sequence 9:09
- Working with Geometric Sequences 5:26
- Finding and Classifying Geometric Sequences 9:17
- How and Why to Use the General Term of a Geometric Sequence 5:14
- The Sum of the First n Terms of a Geometric Sequence 4:57
- Understand the Formula for Infinite Geometric Series 4:41
- How to Calculate a Geometric Series 9:15
- Using Recursive Rules for Arithmetic, Algebraic & Geometric Sequences 5:52
- Arithmetic and Geometric Series: Practice Problems 10:59
- Using Sigma Notation for the Sum of a Series 4:44
- Mathematical Induction: Uses & Proofs 7:48
- How to Find the Value of an Annuity 4:49
- How to Use the Binomial Theorem to Expand a Binomial 8:43
- Special Sequences and How They Are Generated 5:21
- Mathematical Series: Formula & Concept
- Summation Notation: Rules & Examples 3:36
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