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

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

01:03

Prove that if $\sum a_{n}$…

01:16

If $\sum a _ { n }$ conver…

01:09

00:43

00:42

Suppose that $a_{n}>0$ …

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.

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