Srinivasa Ramanujan presented two derivations of “1 + 2 + 3 + 4 + … = −1/12” in chapter 8 of his first notebook. The simpler, less rigorous derivation proceeds in two steps, as follows.
The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + · · ·can be transformed to the alternating series 1 − 2 + 3 − 4 + · · ·. The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century.
In order to transform the series 1 + 2 + 3 + 4 + · · · into 1 − 2 + 3 − 4 + · · ·, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + · · ·, which is 4 times the original series.
These relationships can be expressed with a bit of algebra.
Whatever the “sum” of the series might be, call it c = 1 + 2 + 3 + 4 + …
Then multiply this equation by 4 and subtract the second equation from the first:
The second key insight is that the alternating series 1 − 2 + 3 − 4 + · · · is the formal power series expansion of the function 1/(1 + x)2 with 1 substituted for x.
So it seems that -3c = 1 − 2 + 3 − 4 + · · · = 1/4. Now make another small calculation and voila!
It seems as if c = 1+2+3+4+… = -1/12! (1)
But can this really be true?
Can the sum of POSITIVE numbers equal a negative one?
Can the sum of INTEGERS be a fractional number?
We tend to rely too much on assumptions.
The whole science is based on assumptions.
And when we rely too much on them, we tend to forget they even exist.
Watch closely.
See the “proof” more carefully once more…
Srinivasa Ramanujan presented two derivations of “1 + 2 + 3 + 4 + ⋯ = −1/12” in chapter 8 of his first notebook. The simpler, less rigorous derivation proceeds in two steps, as follows.
The FIRST key insight is that “infinity” exists.
The SECOND key insight is that the series of positive numbers 1 + 2 + 3 + 4 + · · · (we CAN put these dots because of the first assumption) can be transformed to the alternating series 1 − 2 + 3 − 4 + · · ·. The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century.
In order to transform the series 1 + 2 + 3 + 4 + · · · into 1 − 2 + 3 − 4 + · · ·, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + · · ·, which is 4 times the original series.
These relationships can be expressed with a bit of algebra.
Whatever the “sum” of the series might be, call it c = 1 + 2 + 3 + 4 + …
Then multiply this equation by 4 and subtract the second equation from the first:
The second key insight is that the alternating series 1 − 2 + 3 − 4 + · · · is the formal power series expansion of the function 1/(1 + x)2 with 1 substituted for x.
So it seems that -3c = 1 − 2 + 3 − 4 + · · · = 1/4. Now make another small calculation and voila!
It seems as if c = 1+2+3+4+… = -1/12!
But this result is not logical!
So the same process could be actually proof that infinity does not exist!
Infinity…
What is that infinity which we all seek?
What is that infinity to which we all pray?
What made us think about it in a finite cosmos?
Are we humans thinking as Gods?
Or Gods thinking as humans?
Seek the finite.
And you will see the infinity staring back at you…
