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Grandi’s series is divergent. With divergent series, and also conditionally convergent series, you get different “answers” by grouping the terms in different ways.
I agree, Grandi's Series is divergent. And I agree that you get different answers by grouping in different ways.
But please notice I didn't "group the terms" in Grandi's Series as you say. What I did was I lined the terms up for two infinite series (one was Grandi's series, the other was a slightly modified version of the same series). The two series are the same length because they have a simple 1 - 1 correspondence which lasts all the way to infinity.
I think it is quite possible to get different answers like you stated:
- if you group terms or
- if you don't have a 1-1 correspondence (i.e. you skip a term, or clump a a couple of terms or more, or... something) or
- if you stop prematurely (i.e. you convert the infinite series to a finite series),
but I didn't do any of those things.
Could you show me a different answer for Grandi's Series other than 1/2? Of course, it would be best if you used the same technique I used i.e. lining up two infinite series using a demonstrable 1-1 correspondence. It would really help me to understand!
thanks,
John
PS I really don't know if my "proof" is right. I'm not insisting or anything like that.
It's just that I haven't broken any axioms or operational rules in Integer Algebra. As far as I know, each operation I'm using has no flaw. So, to me, the "proof" looks good.
PSS I do realize I am making these assumptions:
1) I assume that it is ok to sum a1 + a2 + a3... to infinity. As someone pointed out, "+" is defined as a binary operation, so it can only take two operands. So a1 + a2 + a3 ... is really a1 + (a2 + (a3 + .... ))))) <-- an infinite number of parentheses here -- which may cause a problem. I don't see it, but maybe there is one.
2) it's ok to sum the terms of two infinite series if they are the same length, and there's no shifting or misalignment issues way down the line, i.e. the two series start aligned and stay aligned to infinity.
3) the sum of 0+0+0... to infinity is 0. Again it seems reasonable, but see #1
4) algebraic manipulations like "2S = 1 --> divide both sides by 2 --> S = 1/2." are ok to do when S started off as an infinite series. I know these manipulations are ok to do if S is a simple, finite number. But it is possible, I guess, that if S actually has the value "infinity" then you do an operation like "2S=1 --> divide both sides by 2", it will result/report as a simple, finite number. It seems very, very unlikely that this would be the case, but I don't know for sure.
5) others?
If you guys see a flaw in any step or operation I've done or any other assumptions, please let me know!
and
Interesting! But there's something a little odd in there ...
Here's my reasoning (not sure if it's true):
- For a finite series, the length of the series (aka cardinality) of "1 to 2n" is different than "1 to 2n+1" for any n you choose.
- But for an infinite series, the length of "1 to 2n" and "1 to 2n+1", as n approaches infinity, is the same (according to Georg Cantor)
- The two infinite lengths are the same since you can draw a 1 - 1 correspondence from each term in the sequence "1 to 2n" to each term in the sequence "1 to 2n + 1" - as long as n goes to infinity! Stopping anywhere along the way breaks the 1 - 1 correspondence, so you can't stop.
- In your two formulas the value you arrive at (0 or 1) is depending on the sequences having different lengths, but in fact both sequences have the same infinite length.
- Another way to look at it, is that n has hit infinity before the "=" sign is reached
I think the correct formulas are:
and
and you can add whatever you want to the summation sign, the limit to infinity overrides it:
John
To me this looks like an easier explanation. Commutative law isn't broken in any way (as far as I can see).
S = 1 - 1 + 1 - 1 + 1 ...
-1 + S = -1 + 1 - 1 + 1 - 1 ...
adding:
S + -1 + S = 0 + 0 + 0 + 0 + 0 ... = 0
-1 + 2S = 0
2S = 1
S = 1/2No worries about having different infinite lengths (sequence "-1 + S" is exactly one longer than sequence S) and so the two sequences line up nicely all the way to infinity. No need to invoke different ways to group the terms at all. And no need to say it has "several values".
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