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Numerade Educator

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Problem 3 Easy Difficulty

Test the series for convergence or divergence.

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

Answer

convergence

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Video Transcript

There's some that we're looking at is alternating sign. So we'd be thinking about the alternating signed test the alternating signed test. You take the absolute value of your terms Case on this case, we're going to get this guy. So then, as long as, Ah, this is eventually decreasing to zero. Then, since we're alternating signs, does this minus one? Then then that means that this sum, this whole thing here, is going to converge. So think of this is a rational function with the the power of the denominator being greater than the power of the numerator. We know that eventually this is goingto be decreasing. Okay? And as we mentioned, since the power, the denominator is greater than the power of the numerator. We know that it'LL approach zero, which is what we want. So this goes to zero as and goes to infinity and as we also commented on property of all rational functions, is that they will eventually be monotone. So the alternating signed test is applicable here and absolute value of our terms in consideration. New approach zero. So we get convergence in this case