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(10 pts) Determine whether the given series converges or diverges_ Be sure justify your answer. (simnilar to 10.6 #18)2 ,(-1y"!...

Question

(10 pts) Determine whether the given series converges or diverges_ Be sure justify your answer. (simnilar to 10.6 #18)2 ,(-1y"!

(10 pts) Determine whether the given series converges or diverges_ Be sure justify your answer. (simnilar to 10.6 #18) 2 ,(-1y"!



Answers

determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ 8+12+18+27+\cdots $$

We have the geometric Siris eight plus 12 plus 18 and so on added all the way up forever. So does this convert or divers in order to determine this we toe look at are the common ratio. We know from the book that if the absolute value of the common ratio is less than one, the Siri's will converge. So let's take a look at our. Our is just one term divided by the previous, and it doesn't matter which two terms we pick as long as they're consecutive. So, for example, we could pick 12/8. This is equal to 6/4, which is equal to 3/2, which is equal to 1.5. Well, that's a problem, because the absolute value of 1.5 is greater than one. So this has failed. Our test has given here. Therefore, this sequence is going to diverge, so there's no sense in trying to sum this up because it just goes all the way out to positive affinity. It does not converge

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.

We have been given on in finite geometric sequence as to minus one, divided by two plus one, divided by eight minus one, divided by 32 plus so on. So the first term is equal to two on the common ratio is equal to minus half, divided by two, which is equal to minus one divided by four now, since the absolute value of far is less than one. So it means that the given sequence is converging, so the some off in terms off such a convergent geometric sequence has given us some mission off. Anyone are raised to the power of K Minus one were key runs from one to infinity, which is given as equal toe, even divided by one minus. R, which will become equal toe two. Divided by one minus minus one. Divided by four, which is equal. Toto divided by five. Divided by four, which will be equal toe eight divided by five


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