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Numerade Educator

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Problem 1 Easy Difficulty

Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^2 - 1}{n^3 + 1} $

Answer

Diverges

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Video Transcript

for this problem. We're going to use the Lim comparison test the and terms those air traditionally the terms that were starting with then the B in terms those the terms that were goingto be comparing our in terms, too, that being term should be somewhat similar to the in terms, in the sense that blame it as an approaches infinity of a n over bian. Hopefully that'LL be something that's non zero and finite. So usually the easiest way to accomplish that is just find the B in terms so that the this ratio that we mentioned is one that's what's happening here. One way to see that is that the AA minus one that we see in the A in terms of the plus one that we see in the end term those that can just be thrown out, so to speak as n goes to infinity as n goes to infinity, the n squared and in Cube they're going to totally dominate and the constants are going to be negligible in comparison. Okay, so that's that's the motivation for construction of this bien. And if you work it out then you'LL see that this limit is indeed one, okay. And what that means is that whatever happens, toa when we sum up, these be in terms. You'LL have the same behavior as if we sum up the's A in terms. So summing up, these be in terms. We're going to be getting the harmonic Siri's when we take in equals one to infinity of one over m the harmonic Siri's diverges, which means that we also must have that summing up the's A in Terms gives us divergence. And again, we note that this one here, the one is not the important part. The important part is that it's something that's non zero and finite that makes the limit comparison test applicable.