Geometric Convergent vs. Divergent - Example 4

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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Video Transcript

So this one says it's gonna be our fourth example out of our convergent versus diversion Siri's. So then the question says, determine whether the following syriza's conversion or divergent and the terms are negative free. Six. Negative 18 and 48 Dada Dada. So from here, remember that because the question is even asking whether it's converging or diverging, we know that it has to be a geometric Siri's because all arithmetic sequences are divergent, so converging means it's coming together. So coming together, whereas diversion means that it's diverging, which means it's going apart. So then let's get our terms s. So we have to first find out the R value of the terms, because that is what's going to determine whether this Siri's converges or diverges. So keep in mind that if it's converging, converging, um, or or if we basically, if it's converging, we know that the R value, the absolute value of the R value, is going to be less than one. Whereas if it's diverging her change, we know that the opposite value of the R value is going to be greater than one. So then, if we look at the terms we have to find the R value which we know can be obtained by taking the G of one or Joseph end whichever term divided by the Jesus End minus one. So the current term divided by the term before. So let's take six and negative three. So six has to go on the numerator because that is the term, um, that comes later on. Then the term that comes before has go on the numerator. That gives me six divided by negative three, which is negative. Two. So we know that going from negative three to negative or positive six, we have to multiply by Negative too. And same, uh, I think that's called this negative, too. So then let's say that that was negative. Two. Which would mean that our our value is negative three and then even going from 60. Negative 18. I can see that if I multiplied by negative three. That becomes negative. 18 Insane. If I do negative 18 times Negative three. That also gives me 48. So I have confirmed that my our value is in fact, negative three. So then if I plugged that into the absolute value, the opposite value of negative three is positive three, which is bigger than one. So that's how we know that our Siri's is diversion. So if we were to kind of plot it out, let me show you what would happen. So let's see. So basically, what I'm gonna do is plot out a situation. Where are term number? It's gonna be So our end is gonna be on our X axis on, then the value is gonna be on the y axis. So for our first term, we had negative too. Second term had positive. Six. Third term. We had negative 18, which is way down there, what city, like way down here. So it's just going to go bigger and bigger and then bigger. So that's kind of what's gonna happen to this, which means it's just gonna keep going like this, get bigger and bigger. So that's why we know it's going to diverge, because it doesn't come towards any one single point