ASSIGNING FINITE VALUES TO INFINITE SERIES
(HOW DOES THE SUM 1+2+3+4+5+… EQUAL -1/12 ? )
Primarily because the electron is thought to be a dimensionless point charge, infinite values of charge density very often appear in the equations of Quantum Field Theory. This results, mathematically, in infinite probabilities being calculated. Infinite probabilities are a problem because probabilities, to be meaningful, must range between 0 and 1, inclusive.
In their calculations, physicists get rid of these infinities by a process they call “renormalization.”
Renormalization involves mathematical manipulations that cancel out infinities, but that leave behind a residue that is contained within those infinities.
One example of this is the claim made by physicists (and mathematicians) that the infinite sum S=n=1∞n = 1+2+3+4+5+… is equal to minus 1/12.
This paper shows the mathematical method that derives this counterintuitive result, and then it presents an analysis of the situation.
The claim
The claim is that 1+2+3+4+5+… = -1/12
The Proof
Here’s how it’s done:
Let S = 1+2+3+4+5+…
Let S1 = 1-1+1-1+1-1+…
Let S2 = 1-2+3-4+5-6+…
Then, summing S2 with itself gives:
S2 = 1-2+3 - 4+ 5- 6+…
+ S2 = + 1- 2 + 3- 4+ 5-…
------
2S2 = 1-1+1 - 1+1 - 1+…
2S2 = S1 equation (1)
Further, notice that:
(1 - S1) = 1 – (1- 1+1- 1+1- 1+…)
= 1 – 1+1- 1+1- 1+1-…
= 0 +1- 1+1- 1+1-…
= 1-1+1- 1+1+…
= S1
Rearranging gives:
2S1 = 1
S1 = ½ equation (2)
Substituting into equation (1) gives:
2S2 = S1
2S2 = ½
S2 = ¼ equation (3)
Now, subtracting S2 from S gives:
S = 1+2+3+4+5 + 6+…
-S2 = - (1- 2+3- 4+5 - 6+…)
S – S2 = 0+4+0+8+0+12+…
= 4 + 8 + 12 + …
= 4 (1 + 2 + 3 + …)
= 4 S
Rearranging gives:
3S = -S2
S = -S2/3
Substituting equation (3) gives:
S = - S2 /3
S = -(1/4)/3
S = - 1/12
Thus it is demonstrated that
S = 1+2+3+4+5+… = - 1/12
The Analysis
A. A simple inspection of S = 1+2+3+4+5+… reveals that this series cannot possibly sum to
– 1/12, for two obvious reasons:
1) Summing positive numbers can never lead to a result that is a negative number, and
2) Summing integers can never lead to a result that is a fraction.
In fact, S = 1+2+3+4+5+… does not have a sum, because it tends to infinity.
Yet, S = 1+2+3+4+5+… = -1/12 was clearly demonstrated above.
B. Given that this series does not have a sum, how in the world did we just demonstrate that its sum is -1/12?
The key step in the mathematics that resulted in that unusual sum is the conclusion, in equation (2) that the series S1 = 1-1+1-1+1-1+… sums to ½. Again, a simple inspection of the series reveals this:
1) As the summation proceeds, beginning on the left, the running sum alternates between two values, and only two values: 1 and 0. The sum never converges to any number. It certainly never has the value of ½.
2) The only way in which the sum could be said to be either 0 or 1 is to cut off the addition at some point. However, cutting off the addition is an illegitimate tactic. When the addition is cut off, the sum is no longer an infinite sum, and it is certainly no longer the original S1.
By inspection, S1 = 1-1+1-1+1-1+… does not have a sum, because it oscillates between 1 and 0.
Yet, S1 = 1-1+1-1+1-1+… = 1/2 was clearly demonstrated above.
C. The logic behind being able to derive such strange fractional values for these obviously infinite (or at least non-convergent) sums is not understood--either by the mathematicians or by the physicists. Yet the procedures shown above are mathematically sound. And, though it is a mystery, those calculated values actually, and surprisingly, work out in practical Quantum Field Theory. Whenever an infinity is encountered in a calculation, it can be made to go away and be replaced by the appropriate finite residue number. This curious method yields calculated results in Quantum Field Theory that actually correspond to the results seen in the laboratory experiments.
There is work to be done by the mathematicians to explain this counterintuitive phenomenon.