The Student Room Group
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>
Competition - post a photo you've taken of nature >>

A question about geometric series

Scroll to see replies

Original post by lizard54142
So you have an "undefined" ratio say. But in order to define a geometric series, you need to know "a", and "r". If you are given these values you can generate any geometric sequence, these values are what defines the sequence. How would you define the sequence 0, 0, 0 .... ? You would define it differently from every other geometric sequence, because r could be any real number. So is it still geometric?

I understand your reasoning, but I am still unconvinced as you can't get away from the issue of dividing by zero. Interesting discussion though :smile:

EDIT: 0, 0, 0 ... is arithmetic in my eyes.

k, k , k, k, ...

doesn't matter who's eyes you have
that is an ARITHMETIC AND GEOMETRIC series for ALL K
Original post by lizard54142
So you have an "undefined" ratio say. But in order to define a geometric series, you need to know "a", and "r". If you are given these values you can generate any geometric sequence, these values are what defines the sequence. How would you define the sequence 0, 0, 0 .... ? You would define it differently from every other geometric sequence, because r could be any real number. So is it still geometric?

I understand your reasoning, but I am still unconvinced as you can't get away from the issue of dividing by zero. Interesting discussion though :smile:

EDIT: 0, 0, 0 ... is arithmetic in my eyes.


But suppose I said that r=0.5, and a=0. Then the sequence is defined so that issue becomes resolved right?
Original post by CancerousNumber
k, k , k, k, ...

doesn't matter who's eyes you have
that is an ARITHMETIC AND GEOMETRIC series for ALL K


r=kkr = \dfrac{k}{k}

let k=0k = 0

r=00r = \dfrac{0}{0}

Where now...
Original post by Xin Xang
But suppose I said that r=0.5, and a=0. Then the sequence is defined so that issue becomes resolved right?


But then let r = 1 and a = 0. Is this the same sequence or a different sequence? Can there be more than one definition of the same sequence?
Original post by lizard54142
r=kkr = \dfrac{k}{k}

let k=0k = 0

r=00r = \dfrac{0}{0}

Where now...


But you are assuming that we should define k first and then r, in which case we can't figure out r. But if you define r first and then choose k=0, you are still left with same sequence, except now its defined.
Original post by Xin Xang
But you are assuming that we should define k first and then r, in which case we can't figure out r. But if you define r first and then choose k=0, you are still left with same sequence, except now its defined.


Original post by lizard54142
But then let r = 1 and a = 0. Is this the same sequence or a different sequence? Can there be more than one definition of the same sequence?


I'll say again what I said here...
Original post by lizard54142
But then let r = 1 and a = 0. Is this the same sequence or a different sequence? Can there be more than one definition of the same sequence?


Why not? Why should sequences be uniquely defined?

Edit: I'm trying to think of examples of sequences that are identical but are defined differently. I understand why I can't find one, but I can't think of a rigorous mathematical explanation. Don't like to hand wave.
(edited 8 years ago)
Original post by lizard54142
r=kkr = \dfrac{k}{k}

let k=0k = 0

r=00r = \dfrac{0}{0}

Where now...

that's a ridiculous argument

k/k = 1 except when k = 0, where k is undefined

but lim k->0 for k/k, by lopitals rule = 1/1 = 1

I have no idea how that is relevant to what I'm saying but whatever
Point to me where it says a sequence that is arithmetic is not allowed to be geometric
Reply 29
Avatar for PrimeLime
PrimeLime
OP
12
Original post by lizard54142
So you have an "undefined" ratio say. But in order to define a geometric series, you need to know "a", and "r". If you are given these values you can generate any geometric sequence, these values are what defines the sequence. How would you define the sequence 0, 0, 0 .... ? You would define it differently from every other geometric sequence, because r could be any real number. So is it still geometric?

I understand your reasoning, but I am still unconvinced as you can't get away from the issue of dividing by zero. Interesting discussion though :smile:

EDIT: 0, 0, 0 ... is arithmetic in my eyes.


I don't think the fact that it is an arithmetic series alone prevents it from being a geometric series.
Original post by lizard54142
But then let r = 1 and a = 0. Is this the same sequence or a different sequence? Can there be more than one definition of the same sequence?

Yes that is the same sequence.

Yes there can be more then one definition of a sequence.

Consider the sequence a=1=r with

U=ar^(3N+2)

V=ar^(N)

Where N is the term number.
Original post by CancerousNumber
Point to me where it says a sequence that is arithmetic is not allowed to be geometric


Original post by PrimeLime
I don't think the fact that it is an arithmetic series alone prevents it from being a geometric series.


Original post by Xin Xang
Yes that is the same sequence.

Yes there can be more then one definition of a sequence.

Consider the sequence a=1=r with

U=ar^(3N+2)

V=ar^(N)

Where N is the term number.


I asked the question on StackExchange (a professional mathematics forum).

http://math.stackexchange.com/questions/1282478/definition-of-a-geometric-sequence

There are divided opinions even here! Thought you might like to watch the discussion.
Reply 32
Avatar for PrimeLime
PrimeLime
OP
12
Original post by lizard54142
I asked the question on StackExchange (a professional mathematics forum).

http://math.stackexchange.com/questions/1282478/definition-of-a-geometric-sequence

There are divided opinions even here! Thought you might like to watch the discussion.


I really like the discussion! Thanks for taking the time to post the question there. :biggrin:
I knew it would be a tricky question to answer!
Reply 33
Avatar for PrimeLime
PrimeLime
OP
12
So I think we've technically solved the problem guys. The result is that we cannot definitively say whether it is a geometric series due to the ambiguity in the definition of a geometric series. I don't like that though. It bothers me that there exists (even a little bit) of ambiguity in the subject that should be completely free of all ambiguity.
Original post by PrimeLime
So I think we've technically solved the problem guys. The result is that we cannot definitively say whether it is a geometric series due to the ambiguity in the definition of a geometric series. I don't like that though. It bothers me that there exists (even a little bit) of ambiguity in the subject that should be completely free of all ambiguity.


Yh it seems as though there are quite a few good ideas there. Quite interesting indeed. May have to sign up soon. That site looks quite interesting.
Reply 35
Avatar for PrimeLime
PrimeLime
OP
12
Original post by Xin Xang
Yh it seems as though there are quite a few good ideas there. Quite interesting indeed. May have to sign up soon. That site looks quite interesting.


I've been there a few times before. They actually put up a lot of really interesting problems there!
Original post by PrimeLime
I've been there a few times before. They actually put up a lot of really interesting problems there!


A lot of the stuff on the site seems a bit beyond me tbh.
Reply 37
Avatar for PrimeLime
PrimeLime
OP
12
Original post by Xin Xang
A lot of the stuff on the site seems a bit beyond me tbh.


Same for me, but occasionally when I'm browsing the internet I come across some good problems on Stack.
Original post by Xin Xang
A lot of the stuff on the site seems a bit beyond me tbh.


Everything on the site is beyond all of us, like I said it is for professionals :smile:
Original post by PrimeLime
I really like the discussion! Thanks for taking the time to post the question there. :biggrin:
I knew it would be a tricky question to answer!


No problem :smile:

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you