Main content

## Algebra 2 (Eureka Math/EngageNY)

### Unit 3: Lesson 12

Topic E: Lesson 29: Geometric series- Summation notation
- Summation notation intro
- Geometric series intro
- Geometric series with sigma notation
- Finite geometric series formula
- Worked example: finite geometric series (sigma notation)
- Worked examples: finite geometric series
- Finite geometric series
- Finite geometric series word problem: social media
- Finite geometric series word problem: mortgage
- Finite geometric series word problems

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Geometric series intro

CCSS.Math:

A geometric series is the sum of the terms of a geometric sequence. Learn more about it here. Created by Sal Khan.

## Want to join the conversation?

- At the end of the video Sal said we'll see if I can actually get a finite value, but how could you get a finite value if you are continuing to infinity?(5 votes)
- Here is a simple example:

∑ 3/10ⁿ over n=1 to ∞

With the first term you get 0.3

When you add the second term you get 0.33

With the third, you get 0.333

As you keep going toward infinity, you get closer and closer to ⅓. So, it is clear this infinite series has a sum of ⅓.(16 votes)

- I'm confused, is a geometric series (or sequence, because i'm still confused about that) a logarithm or exponential function?(5 votes)
**Sequence**: an ordered list of things or terms (numbers). The list can be finite or infinite.**Series**: the sum of the terms in ansequence.**infinte**

We use series to model functions, be they logarithmic functions, exponential functions, trigonometric functions, power functions etc.(4 votes)

- What does that Greek symbol mean?(4 votes)
- This symbol means 'sum'.(1 vote)

- i cant understand

what is the difference between a sequence and series?(3 votes)- A sequence is a collection of objects in a specific order. For example, because integers occur in a particular order, integers comprise a sequence. A sequence may have a finite number of members or it may have infinitely many members.

A series is the sum of all the members of a sequence.(4 votes)

- I am curious if there is any reason why its called a 'geometric series'?(4 votes)
- At1:06Sal says that a (sub n) = 1(1/2)^(n-1). Why is the exponent (n-1)? Why can it not be (n)?(2 votes)
- The first index number of a sequence is n=1.

If we define a_n as 1(1/2)^(n), then the first term of the sequence in the video would be 1(1/2)^(1)= 1/2.

But the first term of the sequence in the video is given as 1.

If we define the sequence as Sal did, then we get 1(1/2)^(n-1) = 1(1/2)^(1-1) = 1(1/2)^0 = 1, as required.(3 votes)

- Sal, at0:35, you wrote {a_n}_n=1.

What my question is, what is the meaning of n=1 there?(3 votes)- This is a compact way of defining a sequence. It means that the sequence terms start from a_1 (indicated by the subscript) and go all the way to infinity (indicated by the superscript). The subscript and superscript following {a_n} are parts of a single definition that indicate lower and upper bounds of "n". So "n=1" that is below should necessarily be considered together with what is above (in this case, "infinity"). Just in case, "infinity" as an upper bound means that the sequence is infinite, i.e. goes on and on forever.(1 vote)

- What is the difference between a geometric series and an arithmetic series?(1 vote)
- In a geometric series, you multiply the 𝑛th term by a certain common ratio 𝑟 in order to get the (𝑛 + 1)th term. In an arithmetic series, you add a common difference 𝑑 to the 𝑛th term in order to get the (𝑛 + 1)th term.(4 votes)

- So from what I've gathered, 'a' is the starting number. "sub" means minus / subtract / take away. And you signify subbing/subtracting (wait is sub an abbreviation?) 'n' by putting it at the bottom rather than next to (like to multiply) or on top (like to square)? If so, why is there no sigma between 'n=1' and '∞', or have 'An' with 'n=1' bellow and '∞' above?(2 votes)
- a can be the starting number, but they usually write a_0. The sub is not an operation like squaring, it is more like declaring which a you are talking about. a sub 4 means the fourth value a will have in the problem.(2 votes)

- ProofWiki and Wikipedia define a series to be the infinite sum of a sequence. But Wikipedia and Sal also say that it's also the limit of the sequence of partial sums of the series - Wikipedia says this is 'by definition'. How can this be by definition - limits of sequences have their own definition - to claim that two definitions are equivalent, you need a proved theorem to connect them ( and I've searched for such a proof but without much success).(1 vote)
- The issue you're having is that there is more than one kind of sum for infinite series.(3 votes)

## Video transcript

Let's say that I have
a geometric series. A geometric sequence,
I should say. We'll talk about
series in a second. So a geometric series,
let's say it starts at 1, and then our common
ratio is 1/2. So the common
ratio is the number that we keep multiplying by. So 1 times 1/2 is 1/2, 1/2
times 1/2 is 1/4, 1/4 times 1/2 is 1/8, and we can keep going
on and on and on forever. This is an infinite
geometric sequence. And we can denote this. We can say that this is equal to
the sequence of a sub n from n equals 1 to infinity, with
a sub n equaling 1 times our common ratio
to the n minus 1. So it's going to be our
first term, which is just 1, times our common
ratio, which is 1/2. 1/2 to the n minus 1. And you can verify it. This right over here you can
view as 1/2 to the 0 power. This is 1/2 half to the first
power, this is 1/2 squared. 1/2 to the first,
this is 1/2 squared. So the first term
is 1/2 to the 0. The second term is 1/2 to the 1. The third term is 1/2 squared. So the nth term is going
to be 1/2 to the n minus 1. So this is just really 1/2
to the n minus 1 power. Fair enough. Now, let's say we
don't just care about looking at the sequence. We actually care about
the sum of the sequence. So we actually
care about not just looking at each of
these terms, see what happens as I keep
multiplying by 1/2, but I actually care
about summing 1 plus 1/2 plus 1/4 plus 1/8, and keep
going on and on and on forever. So this we would now
call a geometric series. And because I keep adding
an infinite number of terms, this is an infinite
geometric series. So this right over here would be
the infinite geometric series. A series you can just view
as the sum of a sequence. Now, how would we denote this? Well, we can use
summing notation. We could say that this
is equal to the sum. We could say that this
is equal to the sum. Let me make sure I'm not
falling off the page. Let me just scroll
over to the left a bit. The sum from n equals 1
to infinity of a sub n. And a sub n is just
1/2 to the n minus 1. 1/2 to the n minus 1 power. So you just say OK,
when n equals 1, it's 1/2 to the 0, which is 1. Then I'm going to
sum that to when n equals 2, which is 1/2,
when n equals 3, it's 1/4. On and on, and on, and on. So all I want to do in this
video is to really clarify differences between
sequences and series, and make you a little bit
comfortable with the notation. In the next few
videos, we'll actually try to take sums
of geometric series and see if we actually
get a finite value.