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Geometric series intro

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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  • spunky sam blue style avatar for user Dr Young
    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)
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    • piceratops ultimate style avatar for user Just Keith
      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)
  • piceratops seed style avatar for user McHenry Walton
    I'm confused, is a geometric series (or sequence, because i'm still confused about that) a logarithm or exponential function?
    (5 votes)
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    • leaf blue style avatar for user Stefen
      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.
      (4 votes)
  • leaf orange style avatar for user YanSu
    What does that Greek symbol mean?
    (4 votes)
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  • leaf yellow style avatar for user Mohd. Nomaan
    i cant understand
    what is the difference between a sequence and series?
    (3 votes)
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    • piceratops ultimate style avatar for user Just Keith
      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)
  • blobby green style avatar for user LetsLearnSomething
    I am curious if there is any reason why its called a 'geometric series'?
    (4 votes)
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  • aqualine seed style avatar for user Guled
    At Sal says that a (sub n) = 1(1/2)^(n-1). Why is the exponent (n-1)? Why can it not be (n)?
    (2 votes)
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    • leaf blue style avatar for user Stefen
      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)
  • duskpin ultimate style avatar for user Sudhanshu Basu Roy
    Sal, at , you wrote {a_n}_n=1.
    What my question is, what is the meaning of n=1 there?
    (3 votes)
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    • purple pi purple style avatar for user Adnan Akhundov
      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)
  • orange juice squid orange style avatar for user Riky M
    What is the difference between a geometric series and an arithmetic series?
    (1 vote)
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    • mr pink red style avatar for user andrewp18
      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)
  • blobby green style avatar for user knuckleheadmusicuk
    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)
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  • aqualine seed style avatar for user Sobhan.Bihan
    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)
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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.