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Numerade Educator



Problem 21 Easy Difficulty

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




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Video Transcript

Let's to note this term here by an and let's try B and equals square root and over n or just one overflow ruin. And here, let's try limit comparison that requires that we look at the limit of a N over Bian as and goes to infinity. So in our problem, what's a over being and our problem? We have a N and then we're dividing by the radical. So that's just square of end times. And so let's just go and multiply our a n by radical And and then here we could go out and simplify this a bit. I can pull an end outside the radical, so first licious factor out end. That leaves us with one plus one over n. And then I could just rewrite this as a product of radicals. So we have ruin times and then, by this term over here ruined and then route one plus one over n and we'LL see shortly the reason for writing it this way. So now let's rewrite this limit to the bottom left. We have limit and to infinity well, we multiply these two radicals together the radicals cancel. We just have an end so and radical one plus one over N over two plus in. And then now we can divide top and bottom by. And so let's go ahead and divide numerator and denominator by the same end. So that'LL cancel those ends there, and then we can rewrite the denominator. It's too over N plus one, and we still have to take the limit. Now let and go to infinity as n goes to infinity one over and into the end. Both code zero. So we just have rather cool one over one, which is just one. And this is good because this any time the limit, let's call it see from the limit comparison test. If it satisfies this inequality, then we can actually use the test. So the chance for our question to see whether the Siri's converges by eleven comparison, we could instead look at the some of the B ends. So let's look at this. I'm here. There's some diverges because if you use the pee test, this is a P. Siri's with P equals one half, because that's the one half power on the radical, and that's the less than or equal to one and any time P is less than or equal to one Europea. Siri's will diverge. So by the limit comparison test, our Siri's also has to diverge and equals one to Infinity Square root one plus end all over two plus in also diverges, and that's our final answer.