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JH
Numerade Educator

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Problem 23 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {5 + 2n}{(1 + n^2)^2} $

Answer

converges

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Video Transcript

you can use either the comparison test here or the limit comparison here. Let's just try using the usual that direct comparison test. So this Siri's right here. I couldn't replace this well first, though, actually, let me just keep it as an equal sign. Let me just go to that denominator first and simplify that. So I just have If I just foil that all out, I have to and square plus one plus into the fourth. Now, I could replace this with First of all, let me make the numerator larger. So I know that five is less than or equal to five in. So I'LL just call this five and plus two one and that's up. So if you make a numerator larger, the fraction is a whole good, larger And not only that, I'LL go ahead and make the denominator even smaller by just keeping the large term into the fourth. So here one plus two and swear plus and fourth is bigger than and to the fourth. So I made my denominator smaller and that also makes the fraction is a hole bigger. Therefore, this is why I'm allowed to just use replaced the previous series with the new series, as long as I'm using this inequality. Now let's go ahead and simplify that and equals one to infinity. That gives us seven n over into the fourth. And then you could go ahead and cross off one of those ends on topping on bottom. Your luck with three over on the bottom, and then we just have seven over in cubed. And now this is a P Siri's. So this converges by the Peters with in our case, P equals three. In any time P is larger than one, you always will have conversions. Therefore, also, we should make another observation that we're dealing with positive terms. Therefore, since we're dealing with positive terms, our series converges. Let's not write our let's actually just right the Siri's and equals one to infinity five plus two in all over one plus and swear square converges by the computer sent test, and that's your final answer