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Numerade Educator

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Problem 2 Easy Difficulty

Test the series for convergence or divergence.

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

Answer

Converges

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Video Transcript

There's another limit comparison problem that's going to be the A in terms that we're using and be in terms We want to be thinking about what happens when n goes to infinity. So n n goes to infinity that minus one and plus one that we see in the in terms they're going to be negligible once we just think about getting rid of those. What we would end up with is ah, one over in squared. So by construction, this being should, you know, look the same as the AI in terms as n goes to infinity. So what we mean by that is that this ratio should be one. As long as this ratio is something that's finite and non zero, then whatever happens when we sum up, all of these be in terms. We should get the same behavior for summing up all of these in terms. Okay, so one is certainly finite in non zero. And if the sum of all of these being terms from n equals one to infinity, we're going to get convergence. And because this ratio is one, we know that summing up all of these in terms from in equals one to infinity. We should also get convergence