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JH
Numerade Educator

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Problem 20 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^2 + n + 1}{n^4 + n^2} $

Answer

Converges

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Video Transcript

let's use the Lim comparison test to show that this Siri's will converge. So here this is my and and then let's define being to be and squared over into the fourth, which I can write is one over and swear now the Siri's converges. Why is that so this converges, since it's a P series with P equals two bigger than one. So here, amusing the Pee test from eleven point three the previous section. So Bea and will converge and then we'LL use limit comparison test. So we want to look at this quantity the limit of a n divided by bien And what we're hoping for is that this number satisfies the inequality, basically sees a positive number, not including zero, and it's less than infinity. So it's a real number that's positive. So let's evaluate this. Go ahead and take this limit. So first, let's divide. So we're doing a N, which is over here and then divide this entire expression by one over and swear, since we're dividing by a fraction, that's just multiplying a and over. Bien is just a end times and squared so good and multiply and swear to this numerator we have a into the fourth power and Cube and square and then into the fourth plus and square. Now let's go ahead and divide top and bottom by the highest power that we see in the denominator, which is into the fourth. So on top and bottom. So we're not changing the fraction here because we're doing it to both numerator and denominator. So we have one one over and won over and square and then on the bottom, one plus one over and square. Now go ahead and take that limit. Let and go to Infinity and the numerator. These two terms go to zero and the denominator that's from Ghost zero and we just have one. So C equals one, satisfies this inequality. So by a limit comparison, our Siri's, which was the sum from n equals one to infinity of a M, also converges. And that's a final answer