Problem 19 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n + 1}{n^3 + n} $

Answer

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Video Transcript

to determine whether the Siri's converges or divergence. So let me call this term here a end and let me also define being two B and O ver and cubed, which I could write is one over and square. And the reason I wrote it in this farm over here is im just looking at the largest term in the new braver and also the larger term in the denominator, and I just simplified that. So here I'm setting myself up to use the Lim comparison test. So first we have to evaluate this Kwan this quantity, Let's call it. See, we have to look at and over Bien. Then if we end up finding out that this number C is larger than zero and it's finite, so it's less than infinity. Then, instead of answering conversions or diversions for our own problem, we could instead compare with the easier problem of one over and swear. And we already know just by looking at it that this Siri's will converge because it's a P series with P equals two. So hopefully when we evaluate this limit, it satisfies this inequality, and then we can use the test. So let's go ahead and fancy and over B end. So not that in your problem. And Overby End is just a and over one over and square and that's just end square And so here licious multiply our a n by and square and I should have written the limit This is out in the front limit is n goes to infinity Now let's go ahead and look inside the numerator Actually just a denominator here and you take the large on our end And then we'll divide It's happened bottom bye and puke So that's one on top and then one on the bottom as well. And when we do this no. So this is all inside of the limit. So then any cubed divided by in Cuba was won and then and squared, divided by in Cuba was won over and and similarly in the denominator, this is what we have now Go ahead and take that limit. Let an approach infinity so that one over and and one over and squared both go to zero and we just have one plus zero over one plus zero and that's equal to one. So c equals one does satisfy the inequality that we need. So we can therefore a Clive Limit comparison test sense the sum of the beings which was won over and swear converges. Why does this one converge? This's just pee test. And then, in your case, P equals two. Therefore, let's not use a semi colon there. Since this happens, this converges by Lim comparison test Our Siri's also conversions. So you're Siri's, which was N plus one over in Pune. Plus end will also converge, and that's your final answer.