Thread: Two pairs of values that have the same AGM

Hi,

I have a pair of known values $\displaystyle (a,b)$ that have the arithmetic geometric mean of $\displaystyle M(a,b)$

I have another pair of values $\displaystyle (c,d)$ which are unknown, but I do know that they have the same arithmetic geometric mean as the first two values, so that $\displaystyle M(a,b)=M(c,d)$.

Is there any way to derive c and d from this info? Or at least to narrow down the possibilities of what c and d might be?

Thanks

I know what an arithmetic mean is and I also know what a geometric mean is, but could you define for me what an arithmetic geometric mean is ?

I learned about it from here:

Arithmetic-geometric mean - Wikipedia, the free encyclopedia

i guess not since saying that the agm are equal is saying that the corresponding sequences have the same limit, but there are multiple different sequences which have the same limits.

for example if you have the sequence from (a,b) then you can find 2 numbers c and d such that they satisfy a_n=g_n for some n. so (c,d) will give rise to a sequence not equal to the first but has the same agm. but there may be many such iterations a_n and g_n.in this case (c,d) would give rise to a subsequence of the first.

Originally Posted by Krahl

i guess not since saying that the agm are equal is saying that the corresponding sequences have the same limit, but there are multiple different sequences which have the same limits.

for example if you have the sequence from (a,b) then you can find 2 numbers c and d such that they satisfy a_n=g_n for some n. so (c,d) will give rise to a sequence not equal to the first but has the same agm. but there may be many such iterations a_n and g_n.in this case (c,d) would give rise to a subsequence of the first.

Hmmm, that's what I was afraid of.

But does a rigorous mathematical proof exist for what you just said?

there is a theorem which says that every subsequence of a sequence converges to the same limit.

Originally Posted by Krahl

there is a theorem which says that every subsequence of a sequence converges to the same limit.

What defines a subsequence? Or what is the relation between a sequence and its subsequence?

from wikipedia ;
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, ABD is a subsequence of ABCDEF.

Originally Posted by Krahl

from wikipedia ;
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, ABD is a subsequence of ABCDEF.

Would you say that every sequence is itself a subsequence of some other sequence, which is itself a subsequence of some other sequence, and so on and so on? Or do there exist sequences which cannot be the subsequence of another sequence?

Every sequence can be extended to another sequence.
eg
(1,1,1,1,1,1,.....)
is a subsequence of
(0,1,1,1,1,1,1,.....)

(........,1,1,1,1,.........)
is a subsequence of
(........,1,1,0,1,1,.........)

note the elements need not be integers