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Consider the infinite series2 (~1)n-1 n5/3 n=1Determine whether the series converges absolutely, conditionally, O not at all.The series converges absolutely: The se...

Question

Consider the infinite series2 (~1)n-1 n5/3 n=1Determine whether the series converges absolutely, conditionally, O not at all.The series converges absolutely: The series converges conditionally: The series diverges.

Consider the infinite series 2 (~1)n-1 n5/3 n=1 Determine whether the series converges absolutely, conditionally, O not at all. The series converges absolutely: The series converges conditionally: The series diverges.



Answers

Determine whether the series converges absolutely, converges conditionally, or diverges. Give reasons for your answers.
$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n !}{2^{n}}$

Okay. And this question a series is given that is submission and equals to 02 in finite, minus one, raised to the power. And it is to the power minus any square. Okay, so there's the series given in the question, and we have to determine whether the series converges absolutely or converges conditionally or diverges. Okay, so first of all, we will do the convergence test for submission. More am. Okay, If submission more n series converges, then we can say it will. It is a case of absolutely convergence. Okay, so now we will go for this. So in discussion and is some sorry cities and submission Any submission and he calls to 12 in finite and minus one race to the power and divided by areas to the power and square. Okay, but we have to check for this. Then submission more than a N will be in submission and equals to one point finite. And it will be one upon areas to the power, any square. Okay. Or we can say it will be submission and equals to one point finite one upon areas to devour end square. Okay, so we get that this is a geometric series with R equals to one by E Okay, and that is always less than one. Therefore, this series is absolutely convergent because this series converges though than our original series will absolutely converges. OK, absolutely can marriages. And this will be our final answer. Thank you.

Okay. And discussion of cities is given that his submission and equals to two in finite minus one Raised to the power and divided by N l N N. Okay, so we can see there is an alternate cities here. And first of all, we will check for submission mode. Am. Okay, If this series converges, then we can say it will be absolutely converges. And if this city diverges, then we will go for the conditional convergence or divergence. Okay, so first of all we will do for that, we can say from the question AM will be okay. More a n will be one upon n l n n Okay and submission. And it was to go to in finite more a n will be submission and equals 222 in finite and one upon and l n n. Okay, so for getting this, we will do. We will do the intrigue away test here, And it will be like interrogation of to 20 night d X divided by X l n x. Okay, we have considerate X l n x okay, and it will be limited. He tends to in finite and integration to t. Okay, and be X over x l n x and now it will be limit He tends to in finite and the integration of this will be a land into the bracket. L n X okay. And the limit is two and over limited T Okay. When we apply the limit, it will be LTD tends to infinite. When we do the parliamentary lower limit, it will be a Len L nt my land. I learned to into the bracket and it okay, and when we apply, that tends to infinite. It will be in finite. Okay, So value of this is in finite. So we can say this is this we can say that integration or so the submission of 22 in finite one upon N l N n and I what it is. Okay, it means submission. More n diverges. Okay, so by this, we can say the answers could not be absolutely convergent. OK, now the answer should be a conditional convergence or divergence. Okay, so for the convergence conditionally convergent, we will check by applying alternative serious test. So, in our question, that is, that is interrogation also a submission and constitute two in finite minus one raised to the power and divided by N. L N N a n is one upon an l n n and for the basic first condition of the alternative serious test that is limited and tends to infinite a n equals to limit and tends to infinite one upon n l n n. And when we apply this, it will be zero. So it's the first condition of the alternative series test for convergence satisfies. Okay. And now for the second condition, that is a n plus one. Okay, so for this, we will do whether it is a decreasing function or not. Okay. For this, a n plus one should be less than a N. It means this is a decreasing function. For this. We will do either derivative. Okay, so let's suppose or function our function. F x is one upon x l n x, then it's derivative. F X will be minus one upon x square l n X minus one upon X square Ln x square Okay, now we can say f dash X will be minus one upon x square l n x okay, and it will be one plus one upon l N X okay. And for all thought all the values of X greater than two f Dash X, that is derivative of FX will always be negative. Okay, Therefore, we can say a n plus one is less than and if this then the second condition also satisfied so that we can say the series converges. Okay, But in the starting, we have mentioned that absolutely convergence cannot be in this in this situation. So the convergence will be conditionally. So our answer will be converges conditionally. Okay. And this will be your final answer. Thank you.

Again. This question of series is given their taste submission and equals to 02 in finite minus one Raised to the power and divided by two n plus one factorial. OK, so in this question we have to determine whether the series absolutely converges, conditionally, converges or diverges. Okay, so in discussion we will check for the series submission More am. Okay, If this series converges, then we can say this series will be converged. Absolutely. Okay, so we will check for this. So as for our caution, submission is submission. And it was 202 in finite minus one, raised to the power and divided by two n plus one Factorial ok, and submission more n will be submission. And it was +202 in finite and it will be won by two and plus one Factorial. OK, so we have to check the convergence of this. So this is not a series of peace. Cities, not any telescopic city is not a geometric series. So we will do Check the convergence of this by the comparison test. Okay, so this is the This is a series of a N and the series of BN will be submission Any cost to zero in finite one upon any square. Okay, so and we know this is the city's, uh this is a P series that is convergent because P equals two here. Okay, so this series converges, so we will compare these cities. Okay, So by the direct comparison, we can say whether direct comparison, we can say the uh huh one upon Sorry, The B n is greater than a n as because one upon two n plus one factorial is less than one upon any square. Okay, so by the direct comparison test, we can say the city this city is also converges. OK, so if submission of more d n series converges, this is the first step. Then we can say it over. Series will be converse. Absolutely. Okay, then. Our answer will be absolutely Convergence converges. Okay. And this will be our final answer. Thank you.

Okay and discussion series is given. That is submission. And it calls to 1.5 night minus one, raised to the power and divided by two days to the power. And okay. And we have to determine whether the series absolutely converges, conditionally, converges or diverges. Okay, so we know it is alternative series, okay? And I'm writing down the tour, um, of the absolutely or conditionally universes. So if the CDs submission more am can work this okay and what it is, and then the and then the cities Submission A n also converges. Okay, converges and it is absolutely I can imagine. Okay, Absolutely convergence. So for absolutely convergence, we should check the convergence of submission of a N. Okay. Here. Submission of more day in here A n s minus one raised to the power one to raise to the government. Okay, so more a n will be one by two by n okay, and submission of more than a N will be submission. And it was 212 in finite one by two. And and now we have to check the convergence of this. So we know it is a geometric series with R equals 21 by two. That is less than one. Therefore, the series converges okay and submission more a n this series converges show. Our original series is absolutely convergent. OK, the answer will be You understand the tour. Um absolutely. Convergence is here. Okay. And that will be our final answer. Thank you.


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