Are convergent and divergent series needed to be seen as different?

[Kyriakos]

A convergent progression is one where the set leads to a specific number, but won't reach it within the progression due to needing infinite steps to go up to that number. Eg 1+1/2+1/4+1/8+... will never reach 2, but it will go ever closer to 2 for as long that it goes on.

A divergent progression is one that goes on to infinity (any type, positive or negative). Eg 1+1+1+1+1... or the natural progression 1+2+3+4+...

But in a way the divergent series has many common points to the convergent one:

-They both lead to something outside of any bounded part of the progressions
-Their end is not particularly relevant as a math object by itself, ie it is not independent of the progression

They differ in the following:

-In a convergent progression the limit is known already. Eg 2 is that limit in the example given of the progression 1+1/2+1/4+... While in a divergent progression the limit is the vague 'infinite', which is not a set number of the kind "2" is largely deemed as.

*

But isn't the limit to infinity again a limit as "2" is? And defined as a limit and not a math concept, isn't infinity there not that different from 2? If you can't ever reach something, and it is proven you cannot reach it, it sort of stops having a particular other quality than the one granting it this non-reachability (?).
Maybe "2" itself is interchangeable with "infinity", much like the limit of an infant in his/her crib are the small bars surrounding the crib, and the limit of movement of any being on earth is the actual planet. Of course the nameless room where the infant looks at the bars of the crib is not the same as the countless locations of our planet. But as a limit is it that different?

 

[Perfection]

"1+1/2+1/4+1/8+... " isn't a set, it's an infinite series, and it is 2 because it represents the limit not the output of in ever increasing finite amount of steps.

 

[Kyriakos]

Thanks. I suspect that is why i called it a series. Should i change also the title or wasn't that obvious enough?

And the topic is if a diverging and a converging series are really different as to the type of formation they are, concerning the notion of a limit.

If you had nothing to write on this why exactly did you deem it cool to post about a typo?

 

[Perfection]

The difference between a set and a series is more than a mere typo. I thought you were conceptually confused as these kinds of problems are typically addressed in a first year calculus course.

 

[Kyriakos]

^Your thought was also obvious, so again why post it? I am sure you do not regard yourself as more intelligent, so you could have just refrained from the trouble

Threads usually ask for discussion, at any rate, not the sort of antagonism you tend to be about. You had posted in other math threads by myself, and i hope you can identify from them if the meaning of set and series is known or not. A set is of things that are defined by a property in common or a lack of that. A series has to feature some sort of progression inherent in it.

 

[Perfection]

Proper terminology is vital for mathematical discourse. It's not antagonistic to ask you to use it. I also think that it will help provide clarity onto your confusions.

This a suggestion made in all sincerity as someone who loves math and is interested in the topic. I'm not interested in beating you over the head on irrelevant details.

 

[BvBPL]

The difference between a set and a series is more than a mere typo.

Thank you for the information. I found that to be helpful.

 

[Kyriakos]

Thank you for the information. I found that to be helpful.

 

[Perfection]

As something more interesting you should note that there actually are series that straddle the border between convergence and divergence. These are called [wiki]Conditionally convergent[/wiki] where the sum depends on the order in which you add the terms.

 

[Zelig]

Yes.

5char

 

[Kyriakos]

As something more interesting you should note that there actually are series that straddle the border between convergence and divergence. These are called [wiki]Conditionally convergent[/wiki] where the sum depends on the order in which you add the terms.

Well, that depends on whether one uses absolutes or not, and using absolutes in series that have both addition and subtraction is obviously going to lead to different results (or in some cases 'all possible results' for a series of such type).

Then again this is not tied to the question in this thread. The series noted are not to be taken as consisting of the absolute values of the numbers in their positions, but as consisting by the values defined with the notation.

A far more relevant example would have been stuff like the fibonacci series, which converges to phi (phi being the value by which any consecutive member in the series is converging to be larger than the immediately previous one) from both directions, ie it gets larger than phi, then smaller than phi, then larger (but less larger) than phi, then smaller (but less smaller) than phi, and so on.

 

[Perfection]

Totally lost dude. What do you mean by "uses absolutes "?

Please do try to get your terminology straight though for clarity's sake. The ratio between successive, fibonacci numbers converging to phi isn't a series but a limit of the ratio between adjacent elements of a sequence.

 

[Perfection]

Anyways maybe this might be more satisfying to ya, k-illa. In standard mathematics an infinite series is the limit of a finite series (a series being the sum of a sequence). To evaluate an infinite series you can find the equation for the sum of a finite series, then take the limit as n "goes to infinity".

But in non-standard analysis intead of taking the limit of equation for the sum a finite series you can plug in a hyperinteger and always get a number back (albeit non-real). For convergent series there will be no hyperreal component (but there will be infinitisimals) and for divergent series there will.

 

[Kyriakos]

Totally lost dude. What do you mean by "uses absolutes "?

Please do try to get your terminology straight though for clarity's sake. The ratio between successive, fibonacci numbers converging to phi isn't a series but a limit of the ratio between adjacent elements of a sequence.

I c. It surely is a massive step from there to construct a series of the ratio, from the original fibonacci series. Surely needed elaboration on that part since it would be way too difficult to gather what was meant. Afterall there definitely could be other stuff meant there and not what you typed as an... more clear/explanatory post?..

As for 'absolute', meh, maybe i mean I -x I = x. Also needs looking into.

Also, do keep in mind that math terms tend to somewhat be different in different languages. I know you are monolingual, but bare it in mind

 

[Perfection]

Of course it's a language barrier issue. I know you mean words other than what you're saying but it's not at all clear what you're actually saying.

Maybe try equations?