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Assume that the power seriesan(r + 4)" converges for €9and diverges for 1 = 2.Which of the following statements about this power series must be true? Div...

Question

Assume that the power seriesan(r + 4)" converges for €9and diverges for 1 = 2.Which of the following statements about this power series must be true? Diverges for I 10_ 2. Diverges for I ~10 Converges absolutely for I 8.9 Converges absolutely for I1and only1and 3 onlyAll must be tnue1,3,and 4 only

Assume that the power series an(r + 4)" converges for € 9and diverges for 1 = 2. Which of the following statements about this power series must be true? Diverges for I 10_ 2. Diverges for I ~10 Converges absolutely for I 8.9 Converges absolutely for I 1and only 1and 3 only All must be tnue 1,3,and 4 only



Answers

Which of the series converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty}(-1)^{n} n^{2}(2 / 3)^{n}$$

Hello. So here we consider the series we have K. Going from one to infinity of K squared plus one over K cube plus one. Well um here we can do education the limit comparison tests again here and choose an appropriate piece series to use for comparison. By examining the behavior of the series for large values of N. So um we're gonna consider here and squared plus one over N cubed plus one. So that's going to be equal to n square times one plus one over and squared divided by and cubed times one plus one over and cube. And then um dividing through here this is going to be equal to one plus one over and squared divided by N. Times one plus one over and cube. So um now that lets us then choose the P series. Okay going from one to infinity of just one over K. So then by the limit comparison test we then take the limit as N tends to infer affinity of one plus one over and squared Times and Times one plus one over and cubed. Um Oops ah times and Over one, which gives us the limit as an tends to infinity of one plus one over and squared over one plus one over and cube and its and goes to infinity. Well that goes to zero so therefore this is equal to 1/1 which is one. So we have to limit here is a positive number and our P series we know the P series one over K is going to diverge. So therefore our given series K squared plus one over K Q plus one. Um By the limit comparison test is also going to die. Virg diverge. Alright, take care.

Considered absolutely convergence. First by looking under submission out and cetera square his dribble. Anybody but to enlist one Victoria, Andi. And you used the limit. Compare on immigration test. Yeah, where any two computer remit and invest to infinity. I am this one. You'll be in this one a toy, a square m streamer And this one leading by endless tree Vittorio, now wanted by the reciprocal will be your endless one Vittorio dividing by and Victoria Square stable and we see the table. And here comes another one about a new front and I would put it about you and you won. There's only 21 cook me a device to get Geico Jenna MIT and goes to infinity. So the first group I have the endless one Antonio Anybody but in Pretoria, Totally square. I have the trees I can bring outside tree here, inside, Ever have the, uh, Theo and this one for Toyo, invented by a chill industry editorial so I can get Geico jittery limit and goes to infinity. But it's when they get equal to Empress One square and terms is when I can put it down here so I can read that in the listing. And this one square, they went in by two endless three times to Angus. Two times with the two and a Swede be done because there s will be out here. So for this, limit him because we have the same decree on the top. And the bottom here therefore, was to get a coach three times. Then go fishing. Next with a maximum barrel would be won over. Jude comes to guide you from there. Forget a gentle phone and smaller than one they've always known Nurseries. There will be absolutely convergent counsel implies conditionally a merchant.

Considered absolutely conversions by look, industries under form and Material square Anybody but you and the tire. And then we will use the symmetry so tensed when we have June. Considered a limit on AM is one of a N who has been an envious one. But during the school year, off your endless too the violin. But I n will be multiplied by the reciprocal too. And Abdullah inviting me and Victoria Square we consume in front is one of them again limit. And just to infinity, we were pulled into the and last one for throwing off over and bacterial very square times that you and Victoria and you find out about you and as Joe Victorio that was signified and we get him it. And just to infinity, this one here will get only equal to end this one square and tense with the is when we get equal job one over to endless, to tempt with it to end this one and we will get the limit. And first infinity nameless one square over two AM just Joe and furniture and less one. So it seemed, because the time in the bottom they had to send a creature. So the limit Nico June, uh, coefficient next to the maximum Bella here. We're being one out of two attempts to before and with it the first moment and one. And for this reason, a man will be absolutely conversion recognized bell conditionally convergent.

Okay and discussion. A series is given their test submission and equals to one point. Finite. Minus one. Raised to the power and plus one divided by end raised to the power four by three. Okay, so there's a series given the question and we have to determine whether the series absolutely converges, conditionally, converges or diverges. Okay, so we will check for the series. More am. Okay, If the series converges, we can say the series, and the question will be converges. Absolutely. Okay, so we will check for this. So as for our question, Ian will be submission. And it was 212 Infinite, actually. Submission again will be this, uh, minus one. Raised to the power and plus one divided by N to the power four by three. Okay. And submission, more am will be mode or sorry, submission and equals to one point finite. It will be one upon and raised to the power four by three. Okay, so we have to check the convergence of this, and we know it is a P series with P equals to four by three. That is greater than one. And when P is greater than one in a P series than the series converges. And if this series converses, then we can say it is a case of absolutely convergent. So what answer will be absolutely converges. Okay, so we're series in, the question will be converges. Absolutely. And that will be our final answer. Thank you.


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