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Summation Notation

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Today we're going to talk about summation notation. To know what summation notation is, we need to go back to what a sequence is, and what a series is. A sequence, if you remember, is just a set of numbers in a particular order. And a series is the sum of the first n terms in a sequence. So then summation notation is just an expression that you can use for the series of a sequence. So today we're going to start by reviewing sequences and series, and then we'll do some examples using summation notation.

So behind me is an example of a sequence. In fact, this is an example of a finite sequence, because it has a limited number of terms. And it's an arithmetic sequence, because it has a common difference of one.

So a series, as I said before, is just the sum of the terms, the first n terms, in a sequence. So if I wanted to find the value of the series for this sequence, I simply would need to add together all of the terms in our sequence. So if we do that, I'll find that 1 plus 2 is 3, 3 plus 3 is 6, 6 plus 4 is 10, and 10 plus 5 gives me 15. So the value of the series for this sequence is 15.

So let's see how we can use summation notation to find the series of more complicated sequences. So let's look more closely at summation notation. Here's an example of a sequence, and this is how we write using summation notation. So the Greek letter sigma here represents the fact that we are finding the sum of the sequence, or the series. And so I'm going to break down each part of the summation notation so you understand what each part means.

So again the sigma indicates that we are adding the terms in our sequence. The an indicates that we are summing the terms in our sequence. N equals 1 is our lower index term, and indicates the first term that we're going to be summing. And above the sigma is our upper index term, and indicates the last term that we're going to be summing. And so our lower index term and our upper index term indicate that we're going to start with the lower index term, and sum up all of the terms up to and including our upper index term. So let's do an example, using summation notation, to find the sum of a sequence.

All right, so if I want to find the summation, or the series of this sequence, I'm going to use my summation notation to write that I want to find the sum of the 1, 2, 3, 4, 5 terms. So I'm going to be summing. My lower index is going to be 1, my first term, and my upper index is going to be 5, because I want to find up to the fifth term.

So using this notation, this is going to become a1 plus a2 plus a3, a4, and a5 . So my n is changing every time, from 1 up till 5, making this a1 up to a5. So now I can just use my values from my sequence. My a1 is 3, a2 is 6, a3 is nine, a4 is 12, and a5 is 15.

So now that I have the terms from my sequence, I can just go ahead and add them together. So 3 plus 6 is going to give me 9. 9 plus 9 is 18. 18 plus 12 is going to give me 30. And 30 plus 15 gives me 45. So using my summation notation, I found that the summation or the series, of the value of the series of this sequence is 45.

So let's look at our key points from today. Make sure that you get them in your notes if you don't have them already, so you can refer to them later. So we started by reviewing the idea of a sequence, which is a set of numbers in a particular order. We then talked about what a series is, and we said that it's the sum of the first n terms in a sequence. We then looked at summation notation, which is just an expression of a series from a lower index term up to and including an upper index term. So I hope that these notes and examples helped you understand a little bit more about summation notation. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.

- Sequence:
- A set of numbers in a particular order.
- Series:
- The sum of the first nth terms in a sequence.
- Summation notation:
- An expression of a series, using the Greek letter sigma, and a lower and upper index to indicate the first and last terms of the sum.

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