Problem 11 Easy Difficulty

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{\sqrt {n^2 + 4}} $

Answer

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Video Transcript

Let's find atleast ten partial sums of the series. So that's s one. For example, we could do the first ten partial sons, so and of course, doing more will only help us. And then we'LL graph the sequence of terms. So that's a one through eight ten. Who are here? A n is given by this formula, and we know by definition of S n that's just the sum of the first and values of a. So now let's go ahead and go to our calculator. So he over and Dez Mel's graphic calculator and we see our formula for and the in term of the sequence is given by the fraction. And then I've plotted the song. The first ten partial sums So capital and is going from zero R Excuse me one to ten, and then I go ahead and compute the partial sums. And here's the graph. The first quarter. This is the end for the end, partial some, and then the right corner is the sum. So this is saying yes. One is approximately point four four seven because of the Pier one comma point for for seven, and similarly we have all the way up to yes, ten. So a poi six for six so we can go ahead and write all the numbers in between the one in the tent and here. If you want to pause the screen here, all the values as one as too, and we can see that they're increasing. And then we have us ten. So we've done to task here. We've found the first ten partial sums approximations, and then we have a graph. This red graph is the graph of the partial sums. So coming back to the problem, we found the first ten partial sons, and we also graft the sequence of partial sums. Now let's go ahead and Graff the sequence of terms. So that's cracking the A's. So let me temporarily just remove the red graph and then let's go ahead and graph and purple there. We have the terms, and if you want to get even, label those. Let's not label. That's not going to help. So here it looks like the ends are getting closer and closer to one. And now let's go ahead and Graff the sequence and the series in the same graph. So purple is a N in the red is Essent, and we can see that the bread graph looks like it's going. You keep increasing all the way up to infinity, whereas the purple graph is getting closer is the one. So now, based on a graph, does it appear that the Siri's converges or diverges based on the graph, it looks diversion? And now if it is that virgin, explain why? Well, in this Siri's diverges bye, they're diversions test. So to show this will just show that the limit as and goes to infinity of a n is not zero. So here I'LL rewrite this denominator. I just pulled and squared out of the radical and I factored it out first. And then I pulled it off the radical. So I know the squared of and square is just end. So this will cancel with that. And then as n goes to infinity, this term goes to zero. So we just have one over radical one equals one, but this is non zero. So by the divergence test, the limit of a N is non zero. That means that the Siri's diverges So that's our final answer