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Geometric Convergent vs. Divergent - Example 2

In mathematics, a convergent sequence is a sequence of real or complex numbers that has a finite limit, i.e. that has a real or complex value that the sequence tends to as the number of terms increases without bound. The terms of a convergent sequence are said to be "converging" to this limit. A sequence that does not converge is called a divergent sequence. The limit of a convergent sequence is also called its sum. The value of a convergent sequence is also called its sum even when the term "sum" is not otherwise used in the context.

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Okay, so this is gonna be the second example out of our convergent versus diversion Siri's and it says, determine whether to following syriza's convergent or divergent. So be sure to pause the video here and try it for yourself. Just toe, actually fully make sure that you understand the process and the concepts of convergent first diversion Siri's So the numbers that were given our 8, 12, 18 and 27. So let's think about how we could get from here to here, here, here and here here. Well, first of all, because we're dealing with conversion versus diversion, all automatically know it has to be a geometric Siri's because arithmetic Siri's air always divergent because they're always going in one direction and the nearly for a geometric Siri's. We know that it can either diverge, meaning it goes in one direction or can converge towards something. Can I ever like this working go something like this something like this towards, like, a zero value, something like that. So in order to find um out whether it's conversion are diversion, we have to look at our value because if they our value is absolute value of the R value is less than a one. That's when we know that it is convergent. If the opposite value of the R values greater than one, that's when we know that it's divergent. So then how do we find the R value? Well, our value is just a ratio between the two terms, so we could take any of those two terms and then draw out of ratio so we can do 8 12, divided by eight. That gives me 1.5. We could also do 27 divided by 18. That also gives me 1.5. We could do 18, divided by 12. That also gives me 1.5. We have confirmed that going from term to term, we're always multiplied by 1.5. So that means that since that 1.5 value is greater than one that are, Siri's is going to be divergent.

Johns Hopkins University
Precalculus

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Heather Z.

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