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Sum of an Infinite Geometric Sequence

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Today, we're going to talk about finding the sum of an infinite geometric sequence. So we'll start by looking at both divergent and convergent sequences, and then we'll do some examples.

So let's start by talking about divergent sequences and series. So I have an example of a divergent sequence, 6, 24, 96, 384, and on and on. And we say this is divergent because the values, as we continue to list on the terms in the sequence, tend toward infinity.

I have another example of a divergent geometric sequence, 8, negative 16, 32, negative 64, and on and on. And this is also an example of a divergent sequence, because even though the terms are alternating between positive and negative, as we continue to list the terms, they're still heading towards the infinite. So we still call it divergent.

And we can say that when the sequence is divergent, then the series or the sum of terms in that sequence are also going to be divergent. And that's when it's an infinite geometric sequence.

And so we can conclude that if an infinite geometric sequence and series are divergent, then that means the absolute value of our r, our common ratio-- the absolute value of r is going to be greater than 1.

So now, let's talk about convergent sequences and series. I've got the infinite geometric sequence 10, 1, 0.1, 0.01, and on and on. We can see that the values of this sequence, as we list out the terms, are tending towards 0. So this is a convergent sequence.

We have another example, 8, negative 4, 2, negative 1, and on and on. And this also is defined as a convergent sequence because even though our values are alternating between positive and negative, they're still tending toward 0. So we still call this a convergent sequence.

And when we're finding the sum of all of the terms in an infinite convergent geometric sequence, the sum is also going to converge to a certain value, because the terms that we will be adding or summing will eventually be virtually 0.

And we can also conclude that in an infinite convergent geometric sequence and series, we have an r value-- the common ratio-- where the absolute value of the r value is going to be less than 1.

So let's look at the formula for the sum of an infinite geometric sequence. But first, let's review the formula for the sum of a finite geometric sequence, which is this formula here. Sn is the sum of the n-terms. a1 is the first term in this sequence. r is the common ratio between any two consecutive terms, and n is the number of terms that we're summing.

So if we are summing an infinite number of terms, such as in an infinite geometric sequence, then our value for n is going to become infinitely large.

And so if our sequence is divergent, as n approaches infinity, r to the n is also going to approach infinity. But if the sequence is convergent as n approaches infinity, then r to the n approaches 0.

So if this value is approaching 0, we can rewrite this formula for an infinite sequence as 1 minus 0 in the numerator, or just 1. And then in the denominator, we'll have 1 minus r.

So this only works for convergent sequences when the absolute value of r is less than 1. And in our formula, instead of writing S sub n, we're simply going to write S, because our value for n is infinite. And so this is our formula for the sum of an infinite geometric sequence.

So let's do an example finding the sum of an infinite geometric sequence using our formula. So I've got an infinite geometric sequence here, 11, 5.5, 2.75, 1.375, et cetera. We need to use our formula to find the sum.

So I'll have S is equal to a1. My first term is 11. 1 minus-- my common ratio r, I can find that by taking any two consecutive terms in my sequence and dividing one term by the one in front of it.

So I'm going to go ahead and take these two terms. So I'll divide 2.75 by the term in front of it, 5.5, which is going to give me 0.5. So my common ratio is 0.5. I'll stick that in my formula.

And I'll start by simplifying in my denominator. 1 minus 0.5 is just 0.50. 1 divided by 0.5 is going to give me 2, and 11 times 2 is 22. So I found that the sum of this infinite geometric sequence is 22. And we also can say that 22 is the value to which this series or the sum of the sequence converges.

So let's go over our key points from today. A sequence with terms whose values tend toward infinity as the sequence continues is defined as divergent. This holds for sequences whose values alternate between positive and negative, but still tend towards the infinities.

When summing all terms in an infinite sequence that is divergent, the sum, or series is also divergent. a sequence with terms whose values tend toward 0 as the sequence continues is defined as convergent. This holds for sequences whose values alternate between positive and negative, but still tend toward 0.

So I hope that these key points and examples helped you understand a little bit more about finding the sum of an infinite geometric sequence. Keep using your notes and keep on practicing, and soon, you'll be a pro. Thanks for watching.

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- Sum of an Infinite Geometric Sequence