If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra (all content)

### Course: Algebra (all content)>Unit 18

Lesson 4: Finite geometric series

# Geometric series with sigma notation

A geometric series is the sum of the terms of a geometric sequence. Learn about geometric series and how they can be written in general terms and using sigma notation. Created by Sal Khan.

## Want to join the conversation?

• Is there a reason why "k" was used for the geometric series notation instead of "i"?
• You can pick any index variable you like. Common choices are i, k, and n.
• Why is the exponent on the r, k? Shouldn't it be (k-1) because the power to which the r is being raised is one less than the term, k, right?
• Value of k for the first term is defined under the sigma.
Since there is k = 0 under the sigma, the value of k in the first term will be 0. Value of k is increased by 1 for every next term.
• In sigma notation is it possible for the index to be a fraction or a decimal? If it is then how does it increase? Does it still increase by 1 ?
• No, the index variable always increases by 1. If you to increase by some other constant, just multiply the index variable that appears in the formula by the necessary number to produce the number you need to increase by.

Suppose you have (x²+7) and need to sum over x=0 to 10 in steps of ½. here is the notation for that:
∑ {(½x)² + 7}, over x=0 to 20
Notice the change in the bounds.
Also note it wouldn't be too hard to simplify this formula before doing the sum.
• Is it valid to take the common factor (here a ) out of the summation notation ?
• Yes. Note that Sum (ar^k) for k = 0 to n = a + ar + ar^2 + ... = a(1 + r + r^2 + ...) so that Sum (ar^k) for k = 0 to n = a*Sum(r^k) for k = 0 to n. When in doubt, expand the series for a bit and see what you can do with normal algebraic manipulation.
• What previous video is Sal talking about? The last video didn't cover geometric series.
• Can 'r' be equal to one?
If so, is 2,2,2,2,2 ..... considered a geometric series ?
• Yes, you could classify it as a geometric series.
• Is it possible for us to convert any geometric series to sigma notation? If yes, please tell me the common method to convert it. If no, why can't we do that?
• I'm pretty sure the answer is yes. Say your first term is a and the common ratio is r. The sigma notation would then be Sum from 0 to n of a(r)^n.
• Is every series written in Sigma notation a geometric series?
(1 vote)
• No. All manner of series can be written using ∑ notation.
• How could one tell whether the sigma notation is geometric or arithmetic?
• The only way to tell is to analyze the formula that appears after the sigma, or the terms, if they are given instead of the formula.
• Would it be okay to make both k's into n's? it seems more confusing to add in the extra variable.
• Hmm, I think it would be even more confusing to try to understand what is meant by a sum that goes from n = 0 to n = n. We need to use k (or something else other than n) as our index variable if we use n as the upper limit in our sigma notation.

## Video transcript

In the last video we saw that a geometric progression, or a geometric sequence, is just a sequence where each successive term is the previous term multiplied by a fixed value. And we call that fixed value the common ratio. So, for example, in this sequence right over here, each term is the previous term multiplied by 2. So 2 is our common ratio. And any non-zero value can be our common ratio. It can even be a negative value. So, for example, you could have a geometric sequence that looks like this. Maybe start at one, and maybe our common ratio, let's say it's negative 3. So 1 times negative 3 is negative 3. Negative 3 times negative 3 is positive 9. Positive 9 times negative 3 is negative 27. And then negative 27 times negative 3 is positive 81. And you could keep going on and on and on. What I now want to focus on in this video is the sum of a geometric progression or a geometric sequence, and we would call that a geometric series. Let's scroll down a little bit. So now we're going to talk about geometric series, which is really just the sum of a geometric sequence. So, for example, a geometric series would just be a sum of this sequence. So if we just said 1 plus negative 3, plus 9, plus negative 27, plus 81, and we were to go on, and on, and on, this would be a geometric series. And we could do it with this one up here just to really make it clear of what we're doing. So if we said 3 plus 6, plus 12, plus 24, plus 48, this once again is a geometric series, just the sum of a geometric sequence or a geometric progression. So how would we represent this in general terms and maybe using sigma notation? Well, we'll start with whatever our first term is. And over here if we want to speak in general terms we could call that a, our first term. So we'll start with our first term, a, and then each successive term that we're going to add is going to be a times our common ratio. And we'll call that common ratio r. So the second term is a times r. Then the third term, we're just going to multiply this one times r. So it's going to be a times r squared. And then we can keep going, plus a times r to the third power. And let's say we're going to do a finite geometric series. So we're not going to just keep on going forever. Let's say we keep going all the way until we get to some a times r to the n. a times r to the n-th power. So how can we represent this with sigma notation? And I encourage you to pause the video and try it on your own. Well, we could think about it this way. And I'll give you a little hint. You could view this term right over here as a times r to the 0. And let me write it down. This is a times r to the 0. This is a times r to the first, r squared, r third, and now the pattern might be emerging for you. So we can write this as the sum, so capital sigma right over here. We can start our index at 0. So we could say from k equals 0 all the way to k equals n of a times r to the k-th power. And so this is, using sigma notation, a general way to represent a geometric series where r is some non-zero common ratio. It can even be a negative value.