Convergent and Divergent series (1 Viewer)

[KeypadSDM]

There are infinitely many of both (I hope), but are there more of one or the other?

As an example, there are infinitely many primes, but there are more composite numbers than primes. Is there a similar relationship with convergent and divergent series?

 

[fitz33]

didnt cantor prove that there are not more composite numbers than primes..?

 

[KeypadSDM]

Logic

Primes can't be even (barring 2), and one isn't a prime. So at least half of all numbers are composite.

9 = 3 * 3, and thus 9 isn't prime.

:. There are more composite numbers than prime numbers. (Under a sufficiently large (N>=9) given limit).

EDIT: But then again, I didn't go to infinity, did I? So that logic doesn't count.

 

[fitz33]

from mathworld:
Any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set. Once one countable set S is given, any other set which can be put into a one-to-one correspondence with S is also countable. Countably infinite sets have cardinal number Aleph-0.


prime numbers can be put in one-to-one correspondence with the natural numbers, e.g.

1 - 2
2 - 3
3 - 5
4 - 7
and so on

 

[fitz33]

but yes, under a sufficiently large limit there are more composite numbers than primes (you left out the "under a limit" part in your 1st post)

 

[turtle_2468]

Short answer: no.
Long answer: No. It all depends on how you define convergent series... there is, after all, no good way to well-define series as there is numbers. (eg you can well-order rationals, but not reals...)

I suppose defining a new series b_n from a_n a convergent series with property b_n=1/a_n could do something. But then you'd have to define b_n=0 if a_n=0, and you need terms tending to 0 if that approach is to work properly...

The fact is, most arbitrary questions in mathematics don't have an answer. Well, a lot of them, anyway. That's because, in most cases (unlike 4U), there may just be no way to figure out the answer, or there IS no answer...

welcome to the real world, keypad (joking!)

 

[KeypadSDM]

Originally posted by KeypadSDM
EDIT: But then again, I didn't go to infinity, did I? So that logic doesn't count.

Sorry guys :S

 

[Affinity]

Originally posted by turtle_2468

I suppose defining a new series b_n from a_n a convergent series with property b_n=1/a_n could do something. But then you'd have to define b_n=0 if a_n=0, and you need terms tending to 0 if that approach is to work properly...

what about b(n) = {1/[a(n) - L]} + L where L is the limit of a(n) then it doesn't have to converge to 0. (every convergent series is then linked to an unique divergent series)

 

[freaking_out]

wow- welcome back affinity!! so what uai and marks did u get?

 

[Affinity]

98 and 97 :S and you?

I am so happy that someone still remembers me, just came back from a long trip

 

[freaking_out]

Originally posted by Affinity
98 and 97 :S and you?

I am so happy that someone still remembers me, just came back from a long trip

woah thats kool- i was expecting 100 from ya. ( and how could we not forget ya, and all the help u gave last year.)

anyways, i was very happy wif my marks- got 93 for both 4u and 3u maths.

btw, what course u doing at uni.?

 

[Affinity]

commerce/science UNSW
probably will make it engineering+commerce if they decide to allow the program

 

[freaking_out]

Originally posted by Affinity
commerce/science UNSW
probably will make it engineering+commerce if they decide to allow the program

bleh- whats wif ppl. and UNSW these days- don't they know that USyd is way better.

 

[Affinity]

you see the problem is, the UNSW student card's quality is much better than the Usyd one

 

[freaking_out]

Originally posted by Affinity
you see the problem is, the UNSW student card's quality is much better than the Usyd one

yes- but what's crap abt. UNSW is the dredded "Bus" that u gotta catch everyday- if i was to go to UNSW, i'll prolly spend more time in the bus than in the train.

 

[BlackJack]

"UAI is inversely proportional to distance from the teacher"
-The Late Syd Adams

On Affinity's sig:
Hooray for distance education and open high school!

And yeah we remember you. Only what ae you doing up at 2.22am? unsw semester is starting... monday...

 

[Affinity]

 

[Grey Council]

you got 98 97 for 3u/4u maths respectively?
holy...

 

[Affinity]

oh.. it's 96 for 3U and 97 for 4U,

 

[freaking_out]

Originally posted by Affinity
oh.. it's 96 for 3U and 97 for 4U,

LoL- u got higher for 4u maths!