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Determine Soru the < Oncoki 1 Soru 1 converges diverges. Socim lemizle converges; find its sum_ 8 Puan1...

Question

Determine Soru the < Oncoki 1 Soru 1 converges diverges. Socim lemizle converges; find its sum_ 8 Puan1

Determine Soru the < Oncoki 1 Soru 1 converges diverges. Socim lemizle converges; find its sum_ 8 Puan 1



Answers

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^3 + 8} $

You were given the falling geometric. Siri's your ass defying what are not converges or diverges. Now, if you're serious coverages, you are asked to find the some. And so you know those the Siri's woke converge. If the following is true, so limit end as a new approaches infinity off eights and is equal to zero. This is when it coverages. Well, as each term goes on of the Siri's, you notice that it is being multiplied 1/8. That means that the denominator is getting larger and larger and larger when the numerator is maintaining out one, so over time is starting to approach zero. So you know that this is true in that this series converges, so to find some you can you large the formula A Over one miles are where a is the first term. Every story, which is 1/8 divided by one minus where r is the mountains be multiplied by every time this is 1/8. So you simplified us too. 1/8, divided by seven overweight, which is equal to 1/8 times 8/7 crosses to out, and you find that some of the Siri's is 1/7

That one. I was honest one window when I was growing up. And this one will be smaller, bigger than this one. And you know that the string one number is good. He's Bye, sergeant. Bye, baby. Siri's fast that vehicle. Half so in his bike, direct from Paris and fast. Bye.

We want to determine if the given series converges. The series is eight plus four plus one half plus 1/4 plus 1/8. Where the repeating pattern or rather the obvious pattern piece. This series is actually in the form of a geometric series for which we can use the steps I listed below to evaluate its convergence. Before we do that, let's define a geometric series is to make sure we understand how to approach this problem property. So a geometric series is always at the for some N equals one to n. A. R to the n minus one or some and equals years with the er to the end. It converges to limit a over one minus are if and only if the value of our is less than one. So let's rewrite this in the form of geometric series and then evaluate whether not it converges and if so the value. So we can really just write this as 12 plus a sum from n equals three of one half times one half to the end, recognizing the pattern. So we have a geometric series with vertical offset 12 equals one half, are equal in half, since absolute value of rss than one. Yes, the series converges and it converges the value 12 plus A over one minus R equals 13.

Threes An equal to 1 to infinity of one of bond And to the power for -7. So here in the execution we have to use the limit comparison tests to determine whether the threes converges or die. Apologies. So here we take is an equal to one upon And to the part 74927 vehicles one upon And to the power 4 -7 we have like run upon And to the ball four as and approaches infinity. So here we take beasts of and equal to one upon and to the power for So now we help limit and approaches infinity of is an upon Visa. And so now we substitute the value of A sub N and B sub N. So here we get limit and approaches infinity of And to the power four upon And to the Power 4 97. So here we have to delight denominator and numerator by and to the powerful. So here we get limit and approaches infinity of one upon one. Now you do seven upon and the power full. So now we can perform direct substitution and approaches infinity. So here we get one. So here see Better than 2 10 for Limit comparison test. So here the c value is one so we can say the limit comparison test applies with sequel to one since threes An equal to 1 to infinity of one upon and to the power for converges. Because here we know that in peace res one upon and to the power p where Be better than 2 1. So sweets will be converges where P less than And equal to one. So suites will be diverges. So here you can see that the value of P is four. So here threes will be converges. The limit comparison test shows that threes and equal to 1 to infinity of one upon and to the power 4 97 also converges. So it is our final


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