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Find the sum of the series 2 +Select one:11Which of the following series are convergent; but not absolutely convergent?Z(-1)"+1 7 + 2 7" + 17 + [Select on...

Question

Find the sum of the series 2 +Select one:11Which of the following series are convergent; but not absolutely convergent?Z(-1)"+1 7 + 2 7" + 17 + [Select one:None

Find the sum of the series 2 + Select one: 1 1 Which of the following series are convergent; but not absolutely convergent? Z(-1)"+1 7 + 2 7" + 1 7 + [ Select one: None



Answers

Determine whether the series is convergent or divergent.
$ \frac {1}{3} + \frac {1}{7} + \frac {1}{11} + \frac {1}{15} + \frac {1}{19} + \cdot \cdot \cdot $

Christian here were given to seize on the form one plus one of the three plus one out of five. Just one out of seven less on and here we can Ready there in the in Judas submission form one on the Jew and a minus one for Andrews from one to infinity. And now, for this reason, we can compare it with this reason I'm the from one of the and and we know that mysteries here will be divergent by the harmonic series. And now I'm going to use the limit comparison test. So I'm Did you computer limit off the where n goes to infinity? We have the one off and dividing by one of a Jew and minus one. Therefore, we get a limit of And just to infinity, off the two a minus one over N on here we will say that this limit with can be written as a limit and just too intimate e there, we're gonna to minus one over n and s. And just to infinity, this one was 20 Therefore, this limit echo choo choo is credit than zero and doesn't implies that by the limit comparison test, we can conclude histories hand will be divergent

Were given a series and were asked to determine whether this series is convergent or divergent. So the series is 1/5 plus 1/7 plus 1/9 plus 1 11 plus 1 13 and so on. When I say and so on, what does this mean? That is, how would you continue find the next term in this some? Well, we see that we'd simply find the next odd number after 13, just 15 and take the reciprocal. So, nurse, we can write this as the sum from and equals, Let's say, zero to infinity of one over five plus to end course. This could also be written as these some from n equals one to infinity of one over two n Plus three doesn't really affect our method either way. Now, let's take ffx be the function 1/2 X plus three. Now we know that F of X is non negative on the interval from one to infinity. Indeed, it's non negative for all positive values of X. We also know that F is mon atomically decreasing on the interval from one to infinity. This is because it's an inverse function. In fact, it's decreasing for all X such that the denominator is non zero and therefore it follows that from the integral test follows that our series converges if and only if the integral from one to infinity of one over two. X plus three. The X converges. The question is, does this interval converge well and to go from one to infinity of 1/2 x plus three d x. This is the same as the limit as T approaches infinity of the integral from one duty of 1/2 x plus three d x, and this is equal to taking anti derivatives limit as T approaches infinity of one half times the natural log of two X plus three from X equals one t and plugging in. This is the limit as T approaches infinity of one half times the natural laws of two t plus three minus one half times the natural log of Just see, that's two times one is two plus three is five. Of course you know that limit This T purchase infinity of the natural log of two T plus three is again infinity. This is infinity minus one half natural log of five. Which of course is just infinity and therefore the integral diverges. And so it follows that these series also diverges by the integral test.

But seriously, I'm the expansion form where we're gonna one out of five plus one. Not I'd just one out of 11. That's one of the four thing. And so and here we can register their engine that come back from I'm the one off. Yeah, we should get in. Uh ah. We'll see that Different agenda five to I B three citizen Best Three, three, three and ah, bless to And And this farm one to infinity. Hey, that's funder, Convergent and divergent. I would compare with those stories on the form one of the end and we know this one is a harmonic series and therefore it will be divergent on now. Well, I'm going to use the limit. Come bear region fast. So they're fine. It's computer limit off now one of and dividing bind one of a three endless to when investing infinity Therefore again, a limit on the n goes to infinity to three endless do over. And so they were eco general limit. He was simplified a gun a three plus two over and and as an just you infinity, this one with a 20 Therefore we get is limited to three and therefore by the limit comparison test here we will conclude industries here it will be divergent

Blue calculus student. We are looking at Chapter 12 Section four, Problem number nine, given a geometric series that we have here in blue where to determine whether it converges or divergence and if it convergence than what does it converge to? So the first thing I'm going to consider here is it actually a geometric Siri's And if so, what's the common ratio? So we can take two nights divided by 2/3 and we will get 1/3. So to get from 2/3 2 to 9 So which smelt by by 1/3. And since it's the same thing here times 1/3 you get to 20 sevens. So we do have a just our stay our start, our first term, my ace of one, our first term is 2/3 and our our common ratio is 1/3. So we do have a geometric Siri's. We know that for a geometric Siri's, we considered the absolute value of our common ratio. If it is less than one than we converge, and if the absolute value of R is greater than or equal to one, then it diverges. So in our particular case, we have that are this equals 1/3 and we know that the absolutely of 1/3 is indeed less than one. Therefore, we can conclude our series converges except see there a little bit on. So there's the first part answered. But then, since the convergence, we want to know what it converges to. In other words, we want to know the sum of this geometric series, and we know the sum of the geometrics. Infinite geometric. Siri's is always a over one minus R. So here we're gonna have a was 2/3 and then over one minus R, which was 1/3. So we have 2/3 over 2/3 which is equal to one, which tells us then that we can conclude the sum of our geometric Siri's, or that our seat geometric Siri's converges to a sum of one. I hope that was helpful


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