Re: 1-1/2+1/3-1/4+1/5-1/6+1/7

On 16 Jan., 21:56, tommy1729 <tommy1...@xxxxxxxxx> wrote:
1-1/2+1/3-1/4+1/5-1/6+ ....

this series can be made to sum to anything.

No, *this* series converges to one and only one value.
However, the convergence is not absolute, so we can find
*another* converging series for any given target value;
then this latter series is related to the one above by having the
same summands in a different order - but order is important
for a series (a series is *not* a sum)


apparently most of you dont understand hilbert's hotel

however you dare too claim that i dont !!

but you are the one confused !

if you understand so well answer this simple question :

N = [0,1,2,3,4,5,...]

Apparently you want to talk about a sequence, not the mere set?


"Square" = [0,1,4,9,16,25,36,...]

notice the " point to point correspondance "

now the set containing N but not "Square" is L

rearrange the set N : containing both L and "Square" elements.

this set contains all non-negative integers and is called T.

T = [0,1,4,9,16,25,...,2,3,5,6,7,8,10,...]

now since the "one to one correspondance" we can count oo long for the squares and thus the limit of our series can skip the non-square elements.

Indeed this is indexed by a different ordinal (2 omega) of the same
cardinality.


still dont believe it ?

than answer the simply question : at what finite position does 2 occur in the set T ?

Sets do not specify positions for their elements. The position of 2 is
of course omega,
then comes 3 at omega+1 etc.


if you answer oo ; thats not finite , and thus you admit i am correct.

omega is not precisely the same as oo.


else im curious what number you will come up with ...

as for Riemann's series theorem : just admit it ; its another series "skipping elements" method , similar and justified by Hilbert's hotel.

"skipping elements" is the only way to choose any value of a series , since if you dont , there is divergence as in :

1-1+1-1+1-1+...

or you use analytic continuations which are actually different series.

or you hide the skipping in a subtle multisection.

the principle is so simple.

you are confused ; not me.

.

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