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Using Sigma Notation for the Sum of a Series

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  • 0:01 Sum of a Series
  • 1:33 Sigma Notation
  • 2:19 Using Sigma Notation
  • 3:32 An Example
  • 4:10 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.

In math, we have a notation for a lot of things. When you need to sum up a series, we also have a notation for that. You will learn about this notation, called Sigma notation, in this video lesson.

Sum of a Series

In math, in addition to numbers by themselves, we also have series of numbers where you have a string of numbers. Some of these strings of numbers end after just a few numbers, while other strings never end. You can have strings of numbers with a pattern as well as random strings.

In this video lesson, we will talk about the strings of numbers that have a pattern. If there is a pattern, then we can write the pattern as a formula. Once we have a formula, we can use our special notation when we want to add up our string of numbers, our series.

For example, say we have 2, 4, 6, ... for our series. We see that we have a pattern. Our pattern is that we add two every time we go to the next number. Our third number is 2 + 2 + 2 or 2 + 2*2. Our fourth number is 2 + 2 + 2 + 2 or 2 + 2*3. We can write this pattern as a formula, such as:

Sigma notation

This formula will give us the numbers in our sequence. For example, the fourth term will be a sub 4 = 2 + (4 - 1)2; a sub 4 stands for the fourth term. So, for our current series of 2, 4, 6, our fourth term will be 2 + (4 - 1) or 3 * 2. So, we have 2 + 6, which is 8. Our fourth term is 8.

Sigma Notation

Since we are dealing with math, we sometimes want to add up our terms to see what kind of totals we get. We are usually interested in the total of a limited number of terms. For example, we might want to know the total of just the first six terms or the first 20 terms. For our pattern of 2, 4, 6, etc, we can add up our first five terms by writing 2 + 4 + 6 + 8 + 10 and then performing these additions.

Of course, with this being math, we like to write things as concisely as possible. We like to substitute symbols for more complicated things to make it easier to work with. For our summation, we have our Sigma notation, which looks like a big sideways M.

Sigma notation

This notation tells us to add up our series.

Using Sigma Notation

Let's look at how to use this notation.

Sigma notation

We notice that our Sigma notation, our sideways M, has an n = 1 on the bottom and a 5 on the top. This is telling us to plug our n value into the formula to the right of the Sigma notation starting with n = 1, then moving onto n = 2, n = 3, all the way up to the top number, n = 5.

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