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Determine if the following series absolutely converges, conditionally converges or diverges: State all tests YOU use and explain your work. (6 Points)(~l)n+1 2n + 3...

Question

Determine if the following series absolutely converges, conditionally converges or diverges: State all tests YOU use and explain your work. (6 Points)(~l)n+1 2n + 3 n

Determine if the following series absolutely converges, conditionally converges or diverges: State all tests YOU use and explain your work. (6 Points) (~l)n+1 2n + 3 n



Answers

Determine whether each series converges (absolutely or conditionally) or diverges. Use any applicable test. $6+2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots$

On in finite geometric sequence given to us has its term as six plus two plus two, divided by three and so on, which means that the first term is equal to six and the common ratio is equal toe, so divided by six, which is equal to one divided by three. Now, since the absolute value off the common ratio is less than one, it means that the given geometric sequence is going toe convert. No, the some off in terms for such a convergence, Siri's has given us some mission even are raised to the part off K minus one were key runs from one to infinity, which becomes equal toe, even divided by one minus on, which will be equal to six, divided by one minus one divided by three, which is equal to six divided by so divided by three, which will be equal to nine

We have the infinite Siri's six plus two plus 2/3 and added on so on forever all the way out, as far as we can go, so does this converge or diverge? To determine this? We need to take a look at the absolute value of the common ratios. There's a property state in the book that if the absolute value that common ratio is less than one, then the Siri's will convert. So let's find the common ratio and see if it is less than one. Well, the common ratio could be found by dividing any term by the previous term. So says to over six. It doesn't matter which two terms we pick because it will be the same for any two consecutive terms. That's what's so useful about common ratio, so to over six is equal to 1/3 and the absolute value of 1/3 is in fact less than one. So we have a convergent Siri's What does they converge to? We'll buy another property in the book. If an infinite series converges, then we know it will converge to a one that is the first term divide by one minus R, where R is the absolute for the common ratio. So the first term was just six. And then we have one minus the common ratio, which was 1/3. All right. We can simplify this. Six states the same one minus 1/3 becomes 2/3. Then we can multiply the three up top and divide by two. That gives us three times six is 18 over to, which is nine. So are Siri's does converge, and it converges to the number nine.

Hello. So here we have the series um where we have one third minus 24 plus three minus 46 and so on. So that's the somewhere we have K. Goes from one to infinity of negative one to the K plus one and then times K over K plus two. So here um what we're gonna do is let a sub K be equal to K over K plus two. And then we get the limit as N goes to infinity of and over M plus two. So that's gonna be the limit as N goes to infinity of and over end times one plus two over end and it's N. Goes to infinity. Um The limit here is going to be equal to one and one is not equal to zero. So um therefore by the test for divergence, our given sequence um or are given series is going to be die Virgin.

So the sequences three power and plus one upon 3, 6, 9, 2, 3. And so what we do for two step is to simplify it. So just take three comments from each each, each digits, so it will be three to the power and since it is a number one in 22 and 23 and two, so on up to last time is and absolutely it is the power and plus one. Finally it will be three upon one into 2 and three and 2 or two. and so for limit except the related funds. And we can now write us related function. Sorry, Here nothing will be it will not be related by any chance. Now, fire consuming. It's very simple. We can find out L N N infinity which is 1, 2. So on to infinity is lower times, infinity is zero. So as N approaches infinity, the value zero. So the seconds will can watch. Yeah.


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