Some more math.

Cisco Qid

Active member
Consider the expression:


(1) s = 1 -1 +1 -1......


This can be divided into two forms, one that sums to 0 and the other that sums to 1 depending on weather the number of terms is even or odd. This can be enumerated by two separate functions, s0, with an odd number of terms and s2, with an even number of terms.


We know that


(2) s0 + s2 = 1 since s0 = 1 and s2 = 0.


The expression (1) is undetermined and can be either s0 or s2.


We have that


(3) s = 1 -1 + 1 -1... = 1 -(1 -1 + 1 -1 ...) = 1 -s'


Where s' = 1 - 1 + 1 - 1... and is not the same as s but rather the complimentary in that it is, s, offset by one. It would a mistake to set ,s', equal to ,s, since both are undetermined. On rearranging (3), we have:


(4) s + s' = 1


Which is similar to (2) with the added difference that (4) does not specify which is s0 and which is s2 but rather that if one is the odd function then the other is the even function.


Of course


This is in response to those who want to assign the value of 1/2 to the series via:


(5) s = 1 -1 +1 -1...


(6) s = 1 -(1 -1 +1...) = 1 -s


=> s = 1/2


This can not be done since the s on the left side of (6) is not the same as the one on the right since they are both separately indeterminate. And in fact, one is the complement of the other.


Also


1/(1-S) = 1 + S + S^2 + S^3 + . . . For |S| < 1


Does not apply for S = -1 Where 1/[1 -(-1)] = 1/2 = 1 - 1 + 1 -1 . . . since |S| must be less than 1.


The result is that the series is never 1/2 but is rather is 1 when ending on an odd number and 0 when ending on an even number of terms - it is therefore divergent.


Another way of seeing this is by taking (4) and dividing by 2 or:


(s+s')/2 = 1/2 - This claiming that the value of 1/2 is not, s, but rather the average of s and s'. That is, what many have been claiming is the value of the series (i.e 1/2) is actually the average of the two separate series.


This leads to the inclination that the sum of all positive natural numbers is equal to -1/12 is not sound when it is derived using the following argument:


s1 = 1 - 1 + 1 - 1 + . . . = 1/2 - already assumed invalid by the above argument


s2 = 1 - 2 +3 - 4 + 5 - 6 + . . .


2*s2 = 1 - 2 + 3 - 4 + 5 - 6 + . . .


+ 1 - 2 + 3 - 4 + 5 + . . .


= 1 - 1 + 1 - 1 + 1 . . . = 1/2 - again invalid


=> s2 = 1/4 - which is also strange for a sum of integers.


s = 1 + 2 + 3 + 4 + 5 + . . . - start with the sum of all integers


s - s2 = 1 + 2 + 3 + 4 + 5 + . . .


-(1 - 2 + 3 - 4 - 5 + . . .) = s - 1/4 - substract s2 or 1/4 from both sides


= 0 + 4 + 0 + 8 + 0 + 12 + . . .


= 4 * ( 1 + 2 + 3 + 4 + 5 + . . . ) = 4 * s = s - 1/4 => s = -1/12


or rather that


1 + 2 + 3 + 4... = -1/12 This is stating that the sum of all positive integers is -1/12.


Conclusion:


This result that the sum of all positive numbers is -1/12 is not logically valid when demonstrated as in this derivation since the initial premise that s1 = 1/2 is not valid.
 

inertia

Super Member
Consider the expression:


(1) s = 1 -1 +1 -1......


This can be divided into two forms, one that sums to 0 and the other that sums to 1 depending on weather the number of terms is even or odd. This can be enumerated by two separate functions, s0, with an odd number of terms and s2, with an even number of terms.


We know that


(2) s0 + s2 = 1 since s0 = 1 and s2 = 0.


The expression (1) is undetermined and can be either s0 or s2.


We have that


(3) s = 1 -1 + 1 -1... = 1 -(1 -1 + 1 -1 ...) = 1 -s'


Where s' = 1 - 1 + 1 - 1... and is not the same as s but rather the complimentary in that it is, s, offset by one. It would a mistake to set ,s', equal to ,s, since both are undetermined. On rearranging (3), we have:


(4) s + s' = 1


Which is similar to (2) with the added difference that (4) does not specify which is s0 and which is s2 but rather that if one is the odd function then the other is the even function.


Of course


This is in response to those who want to assign the value of 1/2 to the series via:


(5) s = 1 -1 +1 -1...


(6) s = 1 -(1 -1 +1...) = 1 -s


=> s = 1/2


This can not be done since the s on the left side of (6) is not the same as the one on the right since they are both separately indeterminate. And in fact, one is the complement of the other.


Also


1/(1-S) = 1 + S + S^2 + S^3 + . . . For |S| < 1


Does not apply for S = -1 Where 1/[1 -(-1)] = 1/2 = 1 - 1 + 1 -1 . . . since |S| must be less than 1.


The result is that the series is never 1/2 but is rather is 1 when ending on an odd number and 0 when ending on an even number of terms - it is therefore divergent.


Another way of seeing this is by taking (4) and dividing by 2 or:


(s+s')/2 = 1/2 - This claiming that the value of 1/2 is not, s, but rather the average of s and s'. That is, what many have been claiming is the value of the series (i.e 1/2) is actually the average of the two separate series.


This leads to the inclination that the sum of all positive natural numbers is equal to -1/12 is not sound when it is derived using the following argument:


s1 = 1 - 1 + 1 - 1 + . . . = 1/2 - already assumed invalid by the above argument


s2 = 1 - 2 +3 - 4 + 5 - 6 + . . .


2*s2 = 1 - 2 + 3 - 4 + 5 - 6 + . . .


+ 1 - 2 + 3 - 4 + 5 + . . .


= 1 - 1 + 1 - 1 + 1 . . . = 1/2 - again invalid


=> s2 = 1/4 - which is also strange for a sum of integers.


s = 1 + 2 + 3 + 4 + 5 + . . . - start with the sum of all integers


s - s2 = 1 + 2 + 3 + 4 + 5 + . . .


-(1 - 2 + 3 - 4 - 5 + . . .) = s - 1/4 - substract s2 or 1/4 from both sides


= 0 + 4 + 0 + 8 + 0 + 12 + . . .


= 4 * ( 1 + 2 + 3 + 4 + 5 + . . . ) = 4 * s = s - 1/4 => s = -1/12


or rather that


1 + 2 + 3 + 4... = -1/12 This is stating that the sum of all positive integers is -1/12.


Conclusion:


This result that the sum of all positive numbers is -1/12 is not logically valid when demonstrated as in this derivation since the initial premise that s1 = 1/2 is not valid.

I believe you posted about this series about a decade ago. ( I may be wrong. )

This is known as the Ramanujan Sum.

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inertia

Super Member
euler_diagrams.png


Euler this, and Euler that, and ...

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Cisco Qid

Active member
I believe you posted about this series about a decade ago. ( I may be wrong. )

This is known as the Ramanujan Sum.

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Yes, that was the time I was chastised for wrongly catetorizing the Riemann zeta function which is another way of reaching -1/12 for the sum of all the integers. That was interesting and showed that I was behind the times in mathematics. But I still remain critical that the sum of all integers is -1/12 even though it is used in String Theory.
 

inertia

Super Member
Yes, that was the time I was chastised for wrongly catetorizing the Riemann zeta function which is another way of reaching -1/12 for the sum of all the integers. That was interesting and showed that I was behind the times in mathematics. But I still remain critical that the sum of all integers is -1/12 even though it is used in String Theory.

An interesting mathematical discussion about Ramanujan's influence in string theory is found ---> here.

Equation 2.2 in the above link is also found in Abramowitz and Stegun's famous book: "Handbook of Mathematical Functions" ( ninth printing ), under "combinational analysis", B. Generating function

Screenshot.jpg
In the very next paragraph the author writes:

Screenshot 2023.jpg

As you can see from visual inspection of equation 2.4 above, it applies to the nth mode.

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