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Numerade Educator



Problem 4 Easy Difficulty

Test the series for convergence or divergence.

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




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Video Transcript

for this problem. We might be tempted to think that just since it's alternating signs that we're going to get convergence. But we need to remember that for convergence toe happen in particular, What we would need to accomplish was that all of these terms approach zero as in goes to Infinity. Okay, so just because we're alternating signs, that doesn't mean that it's good enough for convergence. We would also need for these terms toe approach zero, and we can look here as n goes to infinity. So it's probably easier just work with the absolute value. So if the terms go to zero, then certainly the absolute value goes to zero. So it's it's just a cz good toe to just work with this, and that's it. That's probably easier. Todo can. Now if you look to see what happens as n goes to infinity here, we can just do low pitch house rule, right? So if we take Lin that and approaches infinity and square minus one over and squared plus one, there's a couple things you could do here. One of the things you could do is apply. Look, the house rule and get one okay, So what this means is that these terms aren't even going toe zero. So terms approaching zero is not, ah, sufficient condition for convergence. But certainly if your terms do not approach zero, then you cannot have convergence. So there these terms approaching zero is a necessary but not a sufficient condition. So the fact that these terms approaching zero is not accomplished tells us that this definitely can't be convergence Siri's.