AP®︎/College Calculus BC
Conditional & absolute convergence
"Absolute convergence" means a series will converge even when you take the absolute value of each term, while "Conditional convergence" means the series converges but not absolutely.
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- What is a real world application for conditional vs absolute convergence? Or, more importantly, what IS absolute and conditional convergence? I know it means that a series' absolute value either does or doesn't converge, but what does this reveal about the series? What is the difference between a series that converges but absolutely, and one that does converge absolutely? What is the "condition" for the conditional convergence to converge? Don't both types of series, absolute and conditionally converging series, converge under it's regular definition in that when n=infinity, the sum is finite?(29 votes)
- In a conditionally converging series, the series only converges if it is alternating. For example, the series 1/n diverges, but the series (-1)^n/n converges.In this case, the series converges only under certain conditions.
If a series converges absolutely, it converges even if the series is not alternating. 1/n^2 is a good example.
In other words, a series converges absolutely if it converges when you remove the alternating part, and conditionally if it diverges after you remove the alternating part.
Yes, both sums are finite from n-infinity, but if you remove the alternating part in a conditionally converging series, it will be divergent.
I really don't know if there is any real world application for conditional/absolute convergence.
Hope that my answer was not as jumbled as I think it is :)(26 votes)
- I was most disturbed by reading p 149-150 in Prime Obsession By John Derbyshire (page viewable in Google books if you don't have it) where he shows that the alternating series in the video (which apparently is for Ln2 ) can be rearranged to add up to one half the original un-rearranged series. He doesn't say it but by that logic the rearrangement could be repeated to add up to one quarter the original series. etc.
What is going on here? Something must be flawed in the logic. How can changing the order you add terms up change the answer? How do you decide which way is the correct way, if there is a correct way? My faith in math (and the established methods for determining what an infinite series actually sums to) has been shaken to its foundations.(12 votes)
- That is the nature conditionally convergent series. There is a famous and striking theorem of Riemann, known as the Riemann rearrangement theorem, which says that a conditionally convergent series may be rearranged so as to converge to any desired value, or even to diverge (see, e.g. https://en.wikipedia.org/wiki/Riemann_series_theorem).
I can comfort you with that absolutely convergent series, on the other hand, are more nicely behaved. One can show that if a series converges absolutely, then so do all its rearrangements, and every rearrangement converges to the same value. Hence absolutely convergent series are invariant under rearrangement.
The first time I encountered the Riemann rearrangement theorem, I was completely blown away. It is a prime example of why one must proceed rigorously in doing analysis.(24 votes)
- (silly question)
What would a series be called if it converges when the absolute value is taken, but does not converge normally. Does such a series even exist?(11 votes)
- (In what follows, it is to be understood that summation occurs for all integers
n ≥ 1.)
If the series
∑ |a(n)|converges, we say that the series
∑ a(n)is absolutely convergent. It can be proved that if
∑ |a(n)| converges, i.e., if the series is absolutely convergent, then
∑ a(n)also converges. Hence, absolute convergence implies convergence. What's more, in this case we have the inequality
|∑ a(n)| ≤ ∑ |a(n)|.
It should be noted that there exist series which are convergent, but which are not absolutely convergent.(16 votes)
- At the end of the video, how did Sal know that the series (1/2 )^n+1 converged?(6 votes)
- Geometric Series test. 1/2 = r and since r >-1 and <1 it converges.(6 votes)
- So, would this mean that if there was a series that did not alternate and it was convergent, the series would automatically be absolutely convergent? Because you're taking the absolute value and getting the same series?(3 votes)
- A non-alternating, converging series is considered to be absolutely convergent.(3 votes)
- So if it converges only when the abs is taken would it still be absolutely convergent or conditionally convergent since it is only convergent when you take the abs?(3 votes)
- Great question. There shouldn't be any series that satisfy that condition, actually. A series that converges absolutely will also converge without the absolute value. There's a pretty short and simple proof of that statement here, near the top of the page: http://tutorial.math.lamar.edu/Classes/CalcII/AbsoluteConvergence.aspx . Hope that helps.(3 votes)
- Say you have a series, Σ a_n, and you are not sure whether it diverges or converges conditionally or converges absolutely.
Then, you try the absolute convergence test (ACT):
Σ|a_n|, and you find that Σ|a_n| diverges.
So, my question is:
if you you don't know anything else about Σ a_n, except that
Σ|a_n| diverges, is the ACT inconclusive? Or does it tell us that Σ a_n must converge conditionally? Or does it tell us that Σ a_n must diverge? Just something I was confused about.(2 votes)
- Σ|a_n| diverges, is the ACT inconclusive? Yes
It is not hard to come up with examples for which ACT is inconclusive but the alternating series can converge or diverge.(2 votes)
- Mr. Sal, your courses are the best. I have completed so many of them and am almost done with the calculus BC!. I could really use a video that helps me determine what test to use and when. I know them all but I struggle keeping them all in line and determining when to use which one. for the limits you made this amazing flow chart.(2 votes)
- What if an alternating series doesn't converge? Any specific name to it?(1 vote)
- If it is not convergent then it is divergent.(2 votes)
- Can anyone tell me why 1+2+3+4+....=-1/12?(1 vote)
- It doesn't, and Numberphile's video on it is wrong, as they assume that the sum converges (equals a real number) in the first place.
Mathologer has a nice (although rather long) video explaining it:
- [Voiceover] In the video where we introduced the alternating series test, we in fact used the series, we used the infinite series from n equals one to infinity of negative one, to the n plus one over n. We used this as our example to apply the alternating series test, and we proved that this thing right over here converges. So this series, which is one, minus 1/2, plus 1/3, minus 1/4, and it just keeps going on and on and on forever. We used the alternating series test in that video to prove that it converges. So this thing converges. So this converges by alternating series test. Alternating series test, and if you wanna review that, go watch the video on the alternating series test. Now let's think a little bit about what happens if we were to take the absolute value of each of these terms. So if we were to take the absolute value of each of these terms, so if you were to take the sum from n equals one to infinity of the absolute value of negative one to the n plus one over n, well what is this going to be equal to? Well, this numerator is either gonna be one or negative one, the absolute value of that is always gonna be one, so it's going to be that over. And n is always positive, we're going from one to infinity, so it's just going to be equal to the sum, it's going to be equal to the sum from n equals one to infinity of one over n. And this is just the famous harmonic series. And there's this video that we have, and you should look it up on Khan Academy if you don't believe me, on the famous proof that the harmonic series diverges. So the harmonic series is one plus 1/2, plus 1/3, this thing right over here, this thing right over here diverges. And so when you see a series that converges, but if you were to take the absolute value of each of its terms, and then that diverges, we say that this series converges conditionally. You can say it converges, but you could also say it converges conditionally. And the condition is, I guess you could say, that we're not taking the absolute value of each of the terms. And if something converges when you take the absolute value as well, then you say it converges absolutely. And so let's look at an example of that. If I were to take... This series, let's do a geometric series, that might be fun. Actually I'm using these colors too much, let me use another color. Let's say, let's take the sum from n equals one to infinity of negative 1/2 to the n plus one power. We know this is a geometric series where the absolute value of our common ratio is less than one, we know that this converges. And if we were to take the absolute value of each of these terms, so if you were to take the sum, Let me do that in a different color, just to mix things up a little bit. If you were to take the absolute value of each of these terms, so the absolute value of negative 1/2, to the n plus one power, this is going to be the same thing as the sum, from n equals one to infinity of 1/2 to the n plus one. And here once again, the common ratio, the absolute value of the common ratio is less than one, and we've studied this when we looked at geometric series. This also converges. So when we took the absolute value of the terms, it still converged. So for this one, we can say that this converges absolutely. So we've talked a lot already about convergence or divergence, and that's all been good. And what we're doing in this video is we're introducing a nuance or flavors of convergence. So you can converge, but it might be interesting to say well, would it still converge if we took the absolute value of the terms? If it won't, if you converge, but it doesn't converge when you take the absolute value of the terms, then you say it converges conditionally. If it converges, and it still converges when you take the absolute value of the terms, then we say it converges absolutely. Because even if you take the absolute value of the terms, it converges. Hopefully you find that interesting.