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### Course: AP®︎/College Calculus BC>Unit 10

Lesson 6: Comparison tests for convergence

# Worked example: direct comparison test

Using the direct comparison test to determine that the infinite sum of 1/(2ⁿ+n) converges by comparing it to the infinite sum 1/2ⁿ.

## Want to join the conversation?

• how do you choose which series to compare the given series with? let's say, an = 1/((n^2)+2n). How do you choose which bn to compare with an?
• You need to find a series that is similar in behaviour to the one you are testing, yet simpler and that you know whether it converges or diverges.

In this case, your series `an = 𝚺 1/(n²+2n)` is pretty similar to `𝚺 1/n²` (which is a know convergent series), furthermore, the extra `2n` in the denominator of your series will make each term smaller than the corresponding term of our test series, so the convergence is assured.
• What if in this video we choose to use the harmonic series 1/n ? It is bigger than the firs series. The problem is, it diverges. So, why if I find a series that is bigger than converges, I say that my series converges; even though I know there is some other bigger series that diverges? I get the fact that you need the bigger series to converge to say that my series converges, and vice versa, I need the smaller series to diverge in order to being able to say that my series diverges.
My question is: I find two series that are bigger than my series. One diverges and one converges, or even, the smallest one diverges (even though it's bigger than my series) and the bigger one converges. (Maybe it's not possible and my question it's pointless). But what gives me the assurance that yes, I found ONE that converges, so MINE must converge to? It looks convenient, but not convincing.
• The test can only tell you these two things:
Suppose you have a series An.
If you find a CONVERGENT series Bn such that Bn>=An for all n, then An MUST ALSO CONVERGE.
Suppose you have a series An.
If you find a DIVERGENT series Bn such that Bn<=An for all n, then An MUST ALSO DIVERGE.
• where did he get the if 1>1/2 it converges
• There's all these tests to figure out whether or not a series converges/diverges. How do you get the "intuition" of knowing which test to use?
• practice topic:Direct comparison test

What does the below sentence actually mean?I tried a practice Question n the hint was-

"Because our given series is term-by-term greater than a convergent series, the direct comparison test does not apply.
So the direct comparison test is inconclusive"
• It just means that if we know that the "bigger" series diverges, that information is not enough to determine whether or not the "smaller" series will diverge using the Direct Comparison Test
• What rule/test did he use to determine (1/2)^n converges?
• In order to see the formula that he is referring to you need to rewrite (1/2)^n in the form ar^k. If you remember from an earlier video this then converges to a/(1-r) provided that -1<r<1.
With this in mind you can rewrite (1/2)^n in the form ar^k or 1*(1/2)^k the sum of which is a/(1-r) or 1/(1-1/2) which is 1. Or more simply, if you cover half the distance from where you are to where you want to be, eventually you will get there.
See the first paragraph in this video:Proof of infinite geometric series formula
• what are the conditions i can't use the comparison test ?
• Suppose you have two series with terms a sub n and b sub n respectively. A few condition must be met in order to properly use the comparison test. First, the terms of these series must be positive. Second, a sub n must be less than or equal to b sub n. And finally, when the first two conditions are met, the following comparisons can be used to justify a conclusion regarding convergence and divergence:
(1) If the sum of b sub n converges, then the sum of a sub n converges.
(2) If the sum of a sub n diverges, then the sum of b sub n diverges.

Notice however that the following statements are not justified by the comparison test:
(3) If the sum of a sub n converges, then the sum of b sub n converges.
(4) If the sum of b sub n diverges, then the sum of a sub n also diverges.

• Which test is Sal using at to prove that the b sub n series converges?
• how can i get the value that the series will converge to ?
• The value the series converges to isn't the focus with these tests. It is important to recognize if it is convergent or divergent first.
(1 vote)
• So if i don't do the comparison test and just the limit of that series as n goes to infinity, then it comes out to 0. So if I do it that way,is it still correct to say it converges?
• No, looking at the limit of the terms of the sequence can only ever prove divergence.
(1 vote)

## Video transcript

- [Voiceover] Let's think about the infinite series, so we're going to go from n equals one to infinity, of one over two to the n plus n. And what I want to do is see if we can prove whether this thing converges or diverges. And as you can imagine based on the context of where this video shows up on Khan Academy that maybe we will do it using the comparison test. At any point if you feel like you can kinda take this to the finish line, feel free to pause the video and do so. So in order to kind of figure out or get a sense for this series right over here, it never hurts to kind of expand it out a little bit, so let's do that. So this would be equal to when n equals one this is gonna be one over two to the one plus one, so it's gonna be one over two plus one, it's gonna be one third plus that's n equals one. When n equals two it's gonna be one over two squared, which is four plus two plus one over six. Plus let's see we go to three, n equals one, n equals two, n equals three is gonna be one over two to the third, which is eight plus three is 11. So one over 11, maybe I'll do one more term. Two the fourth power is going to be 16 plus four is 20. Plus one over 20 and obviously we just keep going on and on and on. So it looks, it feels like this thing could converge. All of our terms are positive, but they are getting smaller and smaller quite fast. And if we really look at the behavior of the terms as n gets larger and larger, we see that the two to the n in the denominator will grow much, much faster than the n will. So this kind of behaves like one over two to the n, which is a clue of something that we might be able to use for the comparison test. So let me just write that down. So we have one over, so the infinite series from n equals one to infinity of one over two to the n, and so when n equals one this is going to be equal to one half. When n is equal to two this is going to be equal to one fourth. When n is equal to three this is equal to one eighth. When n is equal to four this is equal to one sixteenth. And we go on and on and on and on. And what's interesting about this is we recognize it. This is a geometric series, so let me be clear. This thing right over here, that is the same thing as the sum of from n equals one to infinity of one half to the n power, just writing it in a different way. And since the absolute value of one half, which is just one half, so because the absolute value of one half is less than one we know that this geometric series converges, we know that it converges And actually we even have formulas for finding the exact sum of or to figure out what it converges to. And so we know this thing converges and we see that actually these two series combined meet all of the constraints we need for the comparison test. So let's go back to what we wrote about the comparison test. So the comparison test, we have two series, all of their terms are greater than or equal to zero. All of these terms are greater than or equal to zero. And then for the corresponding terms in one series, all of them are going to be less than or equal to the corresponding terms in the next one. And so if we look over here, we can consider this one, the magenta series, this is kind of our infinite series of dealing with A sub n and that this right over here is, well I already did it in blue this is kind of the blue series. And notice all their terms are nonnegative and the corresponding terms one half is greater than one third, one fourth is greater than one sixth, one eighth is greater than one eleventh. One over two to the n is always going to be greater than one over two to the n plus n for the n that we care about here. And since we know that this converges, since we know that the larger one converges. It's a geometric series where the common ratio the absolute value of the common ratio is less than one, since we know that the larger series converges therefore, the smaller series or the one where every corresponding term is less than the one in the blue one, that one must also converge. So by the comparison test the series in question must also converge.