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10. (10 pts) Find (with justification) the interval of convergence of the power series:(-i)n+i(x- 1)n+1 n +1...

Question

10. (10 pts) Find (with justification) the interval of convergence of the power series:(-i)n+i(x- 1)n+1 n +1

10. (10 pts) Find (with justification) the interval of convergence of the power series: (-i)n+i(x- 1)n+1 n +1



Answers

Find the interval of convergence.
$$\sum_{n=1}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{2^{n} n}$$

Okay. And the question given function is effects equals two submission off. And it was toe for toe, infinity toe the power and X to the power end live, by the way and what we have to find. We have to find the interval of the conversions for days. We will do the ratio test in discussion and will be pull to the power and X to the power and divide by n and a n plus one will be to to the power and plus one x to the power and plus one divide by and plus one Okay, And now what we have to do, we will do the ratio test The racial test ro equals toe limit and tends to in finite a and plus one divide by a and and now we will simply put the value off a n plus one and a n. And it will be ro equals toe limit and tends to infinity. It will be to to the power and plus one x to the power and plus one divided by and plus one. And now one a on the a n. And it will be goto the power end thought X to the power end and this will be end. Okay, Now we have to solve this and we get row Lim n tends to infinity and it will be two in tow the power and multiply by eggs into X to the power and multiply and divide by and plus one doc toe the power end multiply by X to the power And and now it will solve this like cut total the power and by putting the power and X to the power and by X to the power And and what we get is the rule limit and tends toe infinity It will be cool eggs multiply by and upon and plus one simple and we will put and we get ro equals toe limit And as tothis do X when the blood away On what we can write This one plus one of bone and simple And now we get throw equals two to modoff eggs into limit and tends to infinity and one plus one upon end And we know what will come. It will be row puma Rolex and it will be one glass one upon infinity It means rule equals two two modular eggs in blue, one plus Jiro And now But we get is true equals toe to Modoff X. And now, if Roe is less than one, then the Siri's FX convert that cool modoff X less than one. It means X will be less than one by two and greater than negative. One way to this is over. Final answer. Thank you.

Virgins. Let's take the limit as N approaches infinity here Of this. And so we have negative one actually made or the negative because we're taking out the value anyway. That's fine. We're gonna have X to the two N. Plus three. Oliver two n Plus three. That's what we plug an impulse one in there. That's going to multiply. I'm supposed to put and plus three. Okay. That multiplies by two N plus one over X to the two n plus one here. And so simplifying that out. We've got X squared over one is the value of less than one. That's okay because then that still means that it's going to be between negative one and one here. Right? So being in the case let's test the endpoints so we'll have that X equals negative one. So what we'll have is negative one city. That's negative one to the end, times negative one to the to end plus one and then all that over to them 2, 1 Plus one. And so simplifying that out. We've got negative 123 and plus one Over two and plus one here that's still. So it is going to converge by alternating series. Okay let's take a look at the next one. So it takes equals one. We've got negative one to the end, that's one Over two n Plus one. And so that also converges by alternative series as well. So therefore our answer for the interval of compressions, It's going to be equal to negative one come to one brackets on both.

Okay. We have to find the interval of the convergence and the Given Cities is FX equals two submission and equals to zero toe in finite minus one toe the power and divide by route and square plus one x to the power. And here we will do the ratio test for that and will be minus one to the power and divide by an X squared plus one x to the power end and a and plus one will be minus one toe the power and plus one divide by under route and plus one Squire plus one and two X to the power and plus one, we will simplify it first. Then we will move it A and plus one will be minus one and to minus one to the power tend divided by yeah and bless one Squire plus one and do X and two X to the power. Okay, Now we will have ro equals toe limit and tends to in finite a and plus one divided by a n. Now we will put the value off these two. Okay, So what is the value of a n plus one? It is minus one minus one to the power and upon on the road. And plus when the Squire plus one X and two x to the bourbon. And what is the value off a N? It is minus one to the power and X to the power and divide by under Ruth and the square plus one. Okay, now we will simplify that. What we get Limit intends to. In finite. It will be minus one minus one to the power and X into X to the power and minus one toe the power and next to the Baron. Okay, I will make it simpler. Why? Doing this and plus one Squire plus one. And now it will be ro equals to limit and test too. Infinite will be more attacks outside the limit and limit intends to infinite from this. Tow this from this. Tow this and what we get. Any Squire plus one divide by any square. Plus, I'm sorry. And plus one Squire plus one. Okay. And it No. Yeah, you get. And now we have toe take common and the square from here. And we get ruit was too limit and tends to in finite Okay, mold, X limit. And that's doing finite when we get any square. Okay, this will take this and Squire one plus and my this to end and Squire one plus and minus two square plus and minus two. Okay. And no. Now, from this to this and when we will put an Nichols to invite it there what we get Because to Roy cause to mode ax and limit Intense twin finite. We get one plus zero upon one plus zero plus zero. It means it will be Rubik was two more eggs. It will be one. TARU equals two mode X and we know rule less than one. Then the city's FX converge at mode X. Less than one. It means less than when it means ex converge. Less than one greater than negative one. It will be over in travel off convergence. Thank you.

Okay. We have to find the interval of the car Virgins off the given cities and the given cities is FX equals two submission off an equals to one to infinity minus one body power and X to the power to and plus one divide by end or toe the power. And so we will do the ratio test in racial test. In this question, a innes equals toe minus one to the power end X to the power to and plus one divided by end or toe the power and and yeah, and plus one equals toe minus one to the power and plus one Simply put and plus running against and X to the power to and plus one plus one divided by and plus one to do the power and plus one. And now what we do, we do the ratio, test and ratio test. Roy was too limit and tends to infinity a and bless one upon a And And in this question, Roy was to limit and as to infinity and plus one is expert in this equation. Okay, it will be minus one to the power and I'm solving right now, so it will be my minus one and X to the power to multiply were asked to give our two and plus one again divide by n plus one two and toe devour and divided by a n Yanez minus when to the power and on X to the power to end plus one divided by and don't total de power and okay and moored off this by solving we get Rosie questo limit and stands toe infinity and we get minus or one minus one Body power and eggs Squire multiple ever X to the power to and plus one divided by minus one to the power and X to the power to and plus one dot and toe the power end divide by two and plus one and to live our end. And now we saw that we cut that to live our and by this and by this tow this by this too. This and by solving again, we get Roy equals two limit limit. And as to infinity, it will be X Squire, not one by two, and limit. And as to infinity and upon and plus one Okay. And now we get Roy across to Exit Square by two multiply by limit and tends to infinity. We can write this one upon one, plus one by end. Okay, on. When we put any quest to infinity here and then we get access squired by two. And when we put an equals to infinity air, we get one. Okay, So true equals to access square by do. And now we know that if if Loic if raw less than one then the city's FX converters at point more X let them root to it means the interval. It means we get that radius off the convergence r equals to root toe and interval. And in travel off the convergence is in this X has less than photo and greater than minus roto. And we got that Answer this. Thank you.


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