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Is the following sequence convergent or divergent?If convergent, what is the limit as it approaches infinity. Ifdivergent, state why.sequence = { ln( (6n-1)/(2n+3) ...

Question

Is the following sequence convergent or divergent?If convergent, what is the limit as it approaches infinity. Ifdivergent, state why.sequence = { ln( (6n-1)/(2n+3) ) }

Is the following sequence convergent or divergent? If convergent, what is the limit as it approaches infinity. If divergent, state why. sequence = { ln( (6n-1)/(2n+3) ) }



Answers

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.
$$
a_{n}=\ln \left(\frac{1}{n}\right)
$$

In order to determine convergence or divergence. We'll use the ratio test often because any term that has a factorial will cancel nicely with that test in particular, as we do. So we're going to start with the limit off tune as N goes to infinity of each of the ends will increase its index by one. So we'll change all the ends and plus ones anyone. Then we're going to divide by the original just penetrate in this one first and then we're gonna divide by the original function, which is the same thing, is multiplying by the reciprocal. So end of them comes up top and the negative form to the end. Tons and factorial flips to the bottom. Okay, so at this point, the factorial term is usually the easiest to get to cancel. And if we rewrite this, we could write it as and plus one. And the factorial tells us to multiply by one term less than this. So that would move by by end. And then instead of going any further, we could just stop there, and at that point we would be able to divide the end bacterials out, and, um, we'll go ahead and rewrite the bottom here. So we'll write this, uh, we'll distribute on and plus one toe the end and then multiplied any time. But there too. Um, exponents added together, um, it's best toe support them up is to their products. And then at this point, we would have beads and plus ones that would divide into each other. Um, similar logic over here. We could distribute this and then have another negative one to the power one. And so basically, we were doing that son done just but negative one to the end times negative one to the one. And we've got another cancellation with the negative one to the ends. And that's about all we could do here. So then, at this point, we have this negative one. And again, this is the limit is n goes to infinity. But I'm just leaving that off here. And then everything else is actually to the power end, right? Because this term, what's here but the, um, to the end. And then in the denominator, that's turn here. Well, CNN plus one in the denominator. And then it's worth noting as we take the limit to infinity often anything that involves a fraction with a plus or minus one. It's some form of E. And this whole term here in blue is one over by definition. And then it's negative, since we have multiplied by that negative one there and therefore this limit because this now the ratio test also tells us that we're going to take the absolute value. And so because this isn't between zero and positive one, and this is going to absolutely converge and then that tells us that the original um uh, Siri's is going to have a finite some, Um, and it is not going. Teoh diverge off to infinity, so that's it.

We're gonna apply the route test to check that the Siri's converters or die Burgess. So we have the limit one and ghost affinity off the end route off end to the miners and plus one over. No, these root is the same. That having an exponents one of Berlin. So what we have is the limit. One in Costa infinity, off end to the miners and bless one of it in everything. Divide right, and so we can put them inside the parentheses. So we have in over, in, in one, over in square. Over here. This simplifies toe one. So we have the limit when it goes to infinity of end to the miners, one plus one over and square. Now, when it goes to infinity, these goes to zero. And so we have end to the miners one and when. And ghost infinity that will go to Sierra because he's like having one over and and so that will go to Syria in this is smarter than one always. So we know that the serious comfort

Okay, we're gonna use the other. Maybe Syria test to check if Siri's commercials on my purchase. So first we have to calculate the limit one and host affinity of n factorial over three to the end. Now we can rewrite thesis as the factory Ella's 1234 two in and the bottom. We have three to the end. That's 3 10 Street and three in that influence. So we have We can work these as one or 32 or 33 over three or three, etcetera. Now what we observe is that after these first three tears that we can factor them out outside. That's two overnight. We have all numbers multiply that are all less than one and so beginning that one. So we're multiplying both by things that are getting bigger and bigger, always bigger than one. So that will go to infinity. So the limit goes to infinity and not to see her. So the automated Syria test they lost that the Siri's they purchase

To determine if the sequence is convergent or divergent, we need to look at the behavior of the sequence as an approaches infinity. So we're looking for what happens when we take the limit as N approaches infinity of the sequence here, he's, in this case, we can actually split us up into uh to limits. This is the limit as n approaches infinity of one over and plus the limits as N approaches infinity of line in So now the first limit, this is going towards zero, so we have zero plus, but the second one that we know lawn is an increasing function, so as UN approaches infinity, this is also approaching affinity, so therefore this is a divergent secret.


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