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?

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.

I'm confused, is a geometric series (or sequence, because i'm still confused about that) a logarithm or exponential function?

*Sequence*: an ordered list of things or terms (numbers). The list can be finite or infinite. *Series*: the sum of the terms in an _*infinte*_ sequence. We use series to model functions, be they logarithmic functions, exponential functions, trigonometric functions, power functions etc.

i cant understand what is the difference between a sequence and series?

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.

At 1:06 Sal says that a (sub n) = 1(1/2)^(n-1). Why is the exponent (n-1)? Why can it not be (n)?

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.

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?

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.

What is the difference between a geometric series and an arithmetic series?

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.

Main content

Calculus, all content (2017 edition)

Course: Calculus, all content (2017 edition) > Unit 7

Lesson 4: Finite geometric series

Geometric series intro

Name: Geometric series intro
Uploaded: 2013-02-19T19:56:27Z
Description: A geometric series is the sum of the terms of a geometric sequence. Learn more about it here.

Google Classroom

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

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Dr Young
Posted 8 years ago. Direct link to Dr Young's post “At the end of the video S...”
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?
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(6 votes)
Answer
- Just Keith
  Posted 8 years ago. Direct link to Just Keith's post “Here is a simple example:...”
  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.
  Comment on Just Keith's post “Here is a simple example:...”
  (16 votes)
McHenry Walton
Posted 7 years ago. Direct link to McHenry Walton's post “I'm confused, is a geomet...”
I'm confused, is a geometric series (or sequence, because i'm still confused about that) a logarithm or exponential function?
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(5 votes)
Answer
- Stefen
  Posted 7 years ago. Direct link to Stefen's post “*Sequence*: an ordered li...”
  Sequence: an ordered list of things or terms (numbers). The list can be finite or infinite.
  Series: the sum of the terms in an infinte sequence.
  We use series to model functions, be they logarithmic functions, exponential functions, trigonometric functions, power functions etc.
  Comment on Stefen's post “*Sequence*: an ordered li...”
  (5 votes)
YanSu
Posted 8 years ago. Direct link to YanSu's post “What does that Greek symb...”
What does that Greek symbol mean?
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(4 votes)
Answer
- A K
  Posted 3 months ago. Direct link to A K's post “Sum, so in terms of serie...”
  Sum, so in terms of series you would add everything up.
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  (1 vote)
Mohd. Nomaan
Posted 8 years ago. Direct link to Mohd. Nomaan's post “i cant understand what i...”
i cant understand
what is the difference between a sequence and series?
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(3 votes)
Answer
- Just Keith
  Posted 8 years ago. Direct link to Just Keith's post “A sequence is a collectio...”
  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.
  Comment on Just Keith's post “A sequence is a collectio...”
  (4 votes)
Guled
Posted 7 years ago. Direct link to Guled's post “At 1:06 Sal says that a (...”
At
1:06
Sal says that a (sub n) = 1(1/2)^(n-1). Why is the exponent (n-1)? Why can it not be (n)?
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(2 votes)
Answer
- Stefen
  Posted 7 years ago. Direct link to Stefen's post “The first index number of...”
  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.
  Comment on Stefen's post “The first index number of...”
  (3 votes)
Eyram Klu
Posted 9 years ago. Direct link to Eyram Klu's post “Someone might have alread...”
Someone might have already thought this, but are there no shapes involved in geometric sequences?
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(1 vote)
Answer
Sudhanshu Basu Roy
Posted 7 years ago. Direct link to Sudhanshu Basu Roy's post “Sal, at 0:35, you wrote ...”
Sal, at
0:35
, you wrote {a_n}_n=1.
What my question is, what is the meaning of n=1 there?
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(3 votes)
Answer
- Adnan Akhundov
  Posted 7 years ago. Direct link to Adnan Akhundov's post “This is a compact way of ...”
  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.
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  (1 vote)
LetsLearnSomething
Posted 6 years ago. Direct link to LetsLearnSomething's post “I am curious if there is ...”
I am curious if there is any reason why its called a 'geometric series'?
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(4 votes)
Answer
- andrewp18
  Posted 6 years ago. Direct link to andrewp18's post “https://en.wikipedia.org/...”
  https://en.wikipedia.org/wiki/File:GeometricSquares.svg
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  (0 votes)
Riky M
Posted 6 years ago. Direct link to Riky M's post “What is the difference be...”
What is the difference between a geometric series and an arithmetic series?
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(1 vote)
Answer
- andrewp18
  Posted 6 years ago. Direct link to andrewp18's post “In a geometric series, yo...”
  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.
  Button navigates to signup page
  (4 votes)
knuckleheadmusicuk
Posted 7 years ago. Direct link to knuckleheadmusicuk's post “So from what I've gathere...”
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?
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(2 votes)
Answer
- dantwisler
  Posted 7 years ago. Direct link to dantwisler's post “a can be the starting num...”
  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.
  Comment on dantwisler's post “a can be the starting num...”
  (2 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.