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Question 143n + 1 The series 2 n =1 Sn +3is0A None of these 0 B Convergent by the Ratio Test 0 C Divergent by the Root test 0 D Convergent by the Root Test @ E Conv...

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Question 143n + 1 The series 2 n =1 Sn +3is0A None of these 0 B Convergent by the Ratio Test 0 C Divergent by the Root test 0 D Convergent by the Root Test @ E Convergent by the Alternating series Test

Question 14 3n + 1 The series 2 n =1 Sn +3 is 0A None of these 0 B Convergent by the Ratio Test 0 C Divergent by the Root test 0 D Convergent by the Root Test @ E Convergent by the Alternating series Test



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(a) Decide whether the following series is alternating: \[ \sum_{n=1}^{\infty} \frac{\sin n}{n^{3}} \] (b) Use the comparison test to determine whether the following series converges or diverges: \[ \sum_{n=1}^{\infty}\left|\frac{\sin n}{n^{3}}\right| \] (c) Determine whether the following series converges or diverges: \[ \sum_{n=1}^{\infty} \frac{\sin n}{n^{3}} \]

Three to the Siri's. Let's determine whether the ratio test is inconclusive. Suffer part. Ay, let's look at this one first. This will be our end So we'd like to look at the limit as an goes to infinity. Absolute value a n plus one over n So in our case, here's an plus one and then we'LL divide that by an Now let's go ahead and multiply out This will give us and cubed over and plus one Cute Now, if you like, you could go ahead and use Lopez House rule. In any case, you'LL get one when you evaluate this limit and we know that the ratio test is inconclusive when the limit equals one Since our limit equals one in part a we conclude that the ratio test is inconclusive for the first series And now let's go on to party Now this is our man and we could even go ahead and do a little cancellation to right, this is one over it. So we'LL have the limit n goes to infinity once again and plus one over a n Now the numerator This is one over and plus one and the denominator one over it going to flip that denominator and then multiply it once again? You could use Lopez House rule here and you'LL get a limit of one just like in party and we conclude that the ratio test is inconclusive. Here is well, so that's for a party. Now let's go on to Parsi Here. This is our A So once again we start off with the usual limit and absolute value A and plus one over, eh? So we'LL have n plus one minus one over radical and plus one and divide that by a m So now notice that we can ignore the absolute value If we remove the negative signs we could drop the absolute value So what I'LL do here is just take this on to the next page so I could have some more room. So we're still in party, so have three to the end over radical and plus one and then radical end over three the and minus one. So go ahead and cancel a cz Many threes as you can you could cross off and minus one You'LL still have one three up top and then this limit over here we'LL go to one. So this limit becomes three, which is bigger than one. So Siri's see diverges, and in particular, we see that the ratio test is not inconclusive. Here. That's for party. So it's the first one that we come across to, where the ratio test was not inconclusive and let's see what part visas. So this Siri's was from one to infinity radical, one plus and square in the bottom. So now we look at the limit a n plus one over n. So it's not writing this out. So here's my a n plus one, and divide that by a M. So, as usual, we'LL go ahead and take that second fraction, flip it upside down and multiply it, and we can verify that both of these limits are one. So for the first fraction, if it helps you Khun, rewrite this as and plus one over n one plus one over N, And when you take the limit that goes toe one plus zero and the radical, which is just radical one equals one, and similarly, for the second fraction, you can divide by and square or use low Patel's rule. But this limit will also go to one so that we conclude that the limit goes toe one. And so that the ratios has his inconclusive for party. Hi.

It's our objective here, is to see whether or not each. Any of these series is inconclusive by the ratio test. So let's go ahead and take a look at these here. So if we take the limit as n approaches infinity for the first one Get one over and plus one To the 3rd power Over one over into the 3rd power. So when you simplify the fraction here, Get into the third over and plus one to the third power here, They both have to the 3rd power, which means That this would equal one which is inconclusive. So that means that hey is one of your answers. Okay, let's go ahead and take a look at the next one here. So you have an over to end here. So just in the interest of this base, let me see what I can do here. So I'm gonna have and over And plus one over to to the end plus one multiply by the reciprocal here. So to to the end over. And so we see here that these end up canceling to give you two below and then these go to one when you go to infinity. So that equals one half which is not inconclusive here. So it's less than one. We know that converges so we're good. But this one it's a piece. Okay, Take a look at the next one. So you have. All right, so we got the limit as and approaches infinity of So the opposite value of So I'm gonna ignore the negative there. So we have three to the end Over the square root of n plus one times by the reciprocal of squared event over 3 to the end. Nice one. Right? And so that three and -1 here. So that gives us a positive three. That flips Because that's one thing and it flips over so it's three and these will end up cancelling out here To go to one. This one here again gives us the three on top. So we're left with three is greater than one. We know that their purchase. So that does give us a value that we can conclude with And looking at the last one. So we take the limit as N approaches infinity of this. And so we have the square root of n plus one. Floor one plus plus one squared. That's kind of multiplied by one plus and squared over squared event. Alright, so now take a look at this. We see that all of this since it has the same powers on top and bottom Is going to equal to one. So our answers are a Andy are inconclusive per the ratio test rather

That's why we're looking for where the ratio test would be inconclusive. And so let's take a look at that. So we have on this 11 over and plus one squared Times by their super cool. And she's had squared over one. Take the limit as N approaches infinity here, That equals one. So There is one example. So A is inconclusive here. So all right, look at the next one here. So we have the limit as N approaches infinity of N plus one. 2 to the N Plus one. Substitute for the hand over hand. So that equals one half. So that is an actual value. So be is okay Assistant one. It's convergent, right? It's not equal to one. So our objective here is to look for The ratio test equaling one. And so over here We've got thanks to 3 to the end -1. So that's gonna be a negative three to the end over A scroll event. This one 10 C Squared event over 93. And we're taking the opposite value. So here, in this case Uh 3 to the end by this one. Okay, so that will leave us with uh 1/3 to the N -1. I mean 1/3. -1. So it's just three. Just greater than one. So that's not equal to one there. So I can skip C. Two. It's being see All right. So let's take a look at deep. Okay. Was the last one. So we take the limit as as a participant. Itty of Squared event plus one over one plus and plus one squared times. I had one uh since Christmas one on top over squared event. And so then here we haven't seen powers on top, bottom here total. So then that would just be equal to one. So D. Is and conclusive as well. So our answer to this is A and yeah, are inconclusive preparation tests.

Particularly the series. We need to find the interval the coverages first. So one that has an approaches community here so Gonna do three x Tuesday one Plus 1 Power By the by and plus one multiply by the reciprocal. Instead of dividing by what we started with CNN Over three X -2 to the other power. Take a look at this here. Let me simplify everything. We have to have some value of three experts to It was less than one Things were going from negative one 21 start by adding to everything here so we get one as husband three x just less than three By three days. Are they here? One third assistant X which is us that one. Yeah. So then we test out the endpoints Looking into 1 3rd gives us -1 to the end overhead which converges by Australian serious. Yeah. Mhm. On the other side we have one to the end over in temperatures. Play pieces. So then our interval conversions, it's going to be from I'm negative one third 21. So this is also so this is true for conditional compressions. Sure. If you want absolute convergence then it's from negative one third to one where one there is not included mrs ray, it's absolutely President


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