Question
Finding the Interval of Convergence n!(x c)" 1 .3 .5 (2n
Finding the Interval of Convergence n!(x c)" 1 .3 .5 (2n
Answers
Find the interval of convergence. $$ \sum_{n=1}^{\infty} \frac{(-5)^{n}(x-3)^{n}}{n^{2}} $$
In order for this infinite Siri's to converge, we first must make sure that the limit as n approaches infinity of this expression zero So we can set up the numerator so that the expression will be less than one. The absolute value of this expression will be less than what And by using the second tomb, we can arrange it so that the new movie will evaluate to be fight the power of zero, which will be less than or equal to one. I have said the power of negative end can be rewritten as 1/5 to the power of end and sitting X minus three to be 1/5 will give us the bounds. So evaluating for X minus two equals 1/5 will give you 16 5th and for X minus three equals negative. 1/5 you will get 14 5th. So we can say that the bounds The interval of convergence for this if it's Siri's is equal to two 14, 5th to 16. 5th includes
Going to find the Corolla convergence. Take the limit as N approaches infinity. Playing the ratio tests we have one plus one factorial times X might have seat plus one all that over one touch +35 It's two in -1, two and Plus one. Play that by 1- three times 5. two Venezuelan on top and factorial and X men etc. The end of the bottom there. All right. So simplifying what we have here, you pompous one on top and then times X minus E. Well that divided by two N plus one because these kids with that and so take the limit as and approaches community. What we'll have here is X minus C. Hold her too Less than one. So we've got expert to see what's that? Two therefore negative too less than X messy. Just less than two having C to both sides. We have negative two plus sees less than X which is less than two plus C. Okay, means we have to test those so at X equals negative two plus C. You get and factorial times negative two to the end there. Well that divided by one times 3 times five It's two in -1. So because of that fact exponential to the end there so that word diverge, let's go a lot faster. And then at X equals negative positive plus C. See that we have in factorial to to the end and a similar thing happening there as well. So this would also to average because with the numerator it's going a lot faster. So then our interval of conversions is going to be from I think the two plus C. Two plus C.
Okay. The given cities in this question is effects equals two submission off. And it was toe for toe in finite X to the power and divide by and to the power five. So what we have to do we have to find that in travel of the car virgins, Why ratio test? And it was two x to the power end upon and to the power five and a and plus one will be X to the power and plus one divide by N plus one to the power five. Now we have to do the shortest and the form love. The shortest is raw equals two limit and tends to in finite A and plus one divided by a n. Now we will put the value limit and tends to in finite. This will be X and plus one divided by and plus one toe the power five into and put the power five upon X to the power end. Now we will solve this equation row. It was to limit and tense toe in via Nate, and this will be eggs dot x to the power and divided by divided by and plus one to the power five Lord and to the power five upon X to the power and okay. And now we will cut X to the power and by this extra your end. And now row equals toe limit intends to in finite. And it will be x dot and to the power five upon and plus one toe the power life And after solving again, we get so equals toe. We will put more decks outside the box outside the limit and now limit. And as to infinite, we will rewrite this as one person one upon and to the power five. And now we will put an equals to infinite hair and we get roll equals two more X into one upon one plus zero. That is, law equals two mold eggs. And now, if Roe is less than one, then the city's effects converge at the point. More X less than one. It means the interval of the convergence is X less than one and greater than negative one. And this will be our final Ansett. Thank you
In this problem we need to find the interval of convergence of a given series. Now the given series is submission and is equal to two infinity X -3 to the power and divided by end. Now in order to determine the interval of convergence. First of all, know that the Senate coefficient C M will be won by N. Now, if we replace N B I N plus one we will get C N plus one is equal to one by N plus one. Now let us determine the value of key which is equal to the limit as Endings to infinity. The modelers of CN Plus one divided by the models of CNN. Now, that will be equal to limit, intends to infinity. The models of C N plus one is one by endless one because one day and this one will be positive since it is a natural number and it's it's positive. Similarly CN which is one by and will also be positive and that's the model is of CN will be won by end. So this will be the limit as N tends to infinity and divided by n plus one. In order to determine the limit, let us divide the numerator and the denominator by N will have limited tends to infinity one by one plus one by N. As intends to infinity one by intends to zero. So this will be one by one plus zero, which is one by one, which is equal to one. And hence the radius of convergence is one divided by K, which is one divided by one, which is equal to one. So since this series, Our series as X -3, it is centered at the point I was supposed to three. And so the series will convert for models of x minus three is less than the radius of convergence, which is one. So we can write this as minus one, less the next minus three, less than one. And by adding three throughout this inequality we have minus one plus three which is to let the next minus three plus three which is X Less than 1-plus 3, which is cool. So the series will converge for X. And the range to less than X less than four. Now, we also need to consider the end points of the interval, so at the point X is equal to, the series will become summation, N is equal to two infinity X minus tree. So that will be two minus three to the power and divided by N. So this will be some mentioned and equals to two to infinity minus one to the power and divided by end. Now by the alternating series. Test this series will convert because the N. F term is one by end, The magnitude of the term is one by end. N plus one will be less than a. M. And N tends to zero as intends to infinity. So this series will convert And at the other end point of the interval which is x equals to four. The series becomes a mission and is equal to 2 to infinity. 4 -3. Hold the power and divided by end, which will be submission any close to to to infinity one to the power and divided by end. That will be equal to submission any close to to to infinity one divided by end. Now this can be written as one by two plus one by three plus one by four and so on. And we can write that as minus one plus one plus a half plus one third plus 1/4 and so on. And we write that because this series one plus a half plus one, third plus 1/4 and so on is known to be divergent since it is the harmonic series, so on, adding this minus one, this will still be diverging. So this series will divert. Hence the power series diverges that X is equal to four, so it converges that exceed close to two, and a diversion that exit close to four. So the interval of convergence will be to less very close to X, less very close to four. So this is the required interval of convergence of the power series.