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JH
Numerade Educator

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Problem 37 Easy Difficulty

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \ln \left( \frac {n^2 + 1}{2n^2 + 1} \right) $

Answer

$$\ln \frac{1}{2} \neq 0$$

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Video Transcript

determined whether or not the Siri's convergence and if it is confusion will go ahead and find that some as well. So here this is definitely not geometric There it is possible to try the lot properties here to rewrite this. But before I do that, I will try the with the world calls the test for diversions. So let's go ahead and look at the limit of a N where Anne is just the things that are being added here is and goes to infinity So we just have the limit and goes to infinity natural log and square plus one, two, one squared plus one. So technically, I let me go ahead and yeah, you know what? Let me stick with me. Stick within the values of n So I'm gonna stick with this and then we can go ahead and we know how the log behaves. If you think of the log as being defined over the whole roll numbers, this is continuous function, so we could take the limit and put the limit on the inside. So this is just ln and then if you take the limit of this fraction on the inside, you can use, for example, open house rule. But if we take that limit, we give one half, and this is not equal to zero. Recall that Ellen of one equal zero and otherwise Allen of X is not equal to zero if X is not equal to one. And therefore what happens is when we've showed is that we've shown that limit of Ellen these air two conditions that need you need to show divergence. We showed that it exist. And then, too, we showed That's a limit and squared. Plus one No. Two, one squared plus one is not equal to zero. Therefore, this Siri's diverges by the and then I'LL just by the test for divergence. That's the reason for which the syriza emerges, and that's our final answer.