Problem 39 Hard Difficulty

Prove that if $ a_n \ge 0 $ and $ \sum a_n $ converges, then $ \sum a_n^2 $ also converges.

Answer

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Video Transcript

Let's suppose Anne is bigger than or equal to zero and that there's some converges. Now we'd like to show that this I'm also convergence. So by the way, diversions test. So we're using the fact here that there's some convergence we must have that the limit of an goes to zero. So this means there exist some natural number end such that so that there should be such that so may come back and fix that such that if we take a little end to be beyond this capital in then, for example, a N is less than one sent. In other words, since the ANS heir eventually getting close to zero, there becomes a point in which they're eventually less than one. So let's use this fact to work on this some here. So let's sum from one to infinity. It doesn't matter what the starting point is, but since they didn't write one here, let's just go ahead and start at one. Now let me rewrite this sum by just pulling out the first in terms. And then here's the remainder from N plus one all the way to infinity. So we have two sums here now This is finite, since it's just a finite sum. So we don't This will automatically converge now because we have our little and values are bigger than capital and here, because endless one and anything larger than n plus one is bigger than capital. And so at this point, we know that little and bigger than big in implies A M is less than one. That was our assumption over here. So then we also have a and squared less than a N so that it's also less than one for. In that case, I don't even need the one. Any time you take a number that's between zero and one, and if you square it, it will either stay the same size or only it's smaller. Won't get larger if you If you want a more convincing argument, here's the graph Y equals X. We're just and there and then here's a and swear the proble is below as long as you're between zero and one. So this means that in this larger sum here we have less than or equal to. So the next step is to use this upper bound. So we have that first finite sum and then we can replace in the larger sum and then oops, so that should not be squared. That's the first power. And that's by using this fact over here. That's why we have just a in here with the upper bound. Now we know this. Some circled and green converges because that's given information. Therefore, we have a finite sum, and we're adding that to a convergence. Um, so that means that the circle green part is just a real number. In any time you add two real numbers, you just get it back, a real number. So this sum is, ah, whole convergence. So now we can use the comparison test comparison. Dirham, We started off with the series in question here, and we just pounded it above by a conversion. Siri's so by the computer, suggest what the Siri's conversions, and that's your final answer.