Question 1

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?

Accepted Answer

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 :)

Question 2

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.

Accepted Answer

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.

Question 3

(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?

Accepted Answer

(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.

Question 4

At the end of the video, how did Sal know that the series (1/2 )^n+1  converged?

Accepted Answer

Geometric Series test. 1/2 = r and since r >-1 and <1 it converges.

Question 5

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?

Accepted Answer

A non-alternating, converging series is considered to be absolutely convergent.

Question 6

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?

Accepted Answer

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.

Question 7

Can anyone tell me why 1+2+3+4+....=-1/12?

Accepted Answer

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:
https://www.youtube.com/watch?v=YuIIjLr6vUA

Question 8

What if an alternating series doesn't converge? Any specific name to it?

Accepted Answer

If it is not convergent then it is divergent.

Question 9

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.

Accepted Answer

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.

Course: Integral Calculus > Unit 5

Conditional & absolute convergence

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