The Sum of All Natural Numbers?
1 + 2 + 3 + 4 + 5 … = −1
12⁄
By Theo Macklin (Y11)
This succinct, self-confident statement is a reason to hate Euler. Unfortunately, the maths that leads us to this arrogant but strangely compelling summation as simple as it comes. The idea that the infinite sum of positive whole numbers should give a negative fraction is, frankly, unimaginable; however, once again, the sly, mischievous talons of infinity delight in ruining the fundamental bedrock of nursery addition. Despite this initital mental impasse, there are in fact two ways to prove that the sum of positive integers does reach that ridiculous -1/12. There is the scary sounding proof by Leonhard Euler that utilises zeta function regularisation and the more accessible proof by Srinivasa Ramanujan. It should be obvious which one I intend to start with.
Ramanujan’s Proof:
Ramanujan’s proof requires two realisations: these require us to utilise algebra to represent the constants of series, let’s call them C, C 1 and C 2 . The first is such:
1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 … = 1 2⁄
This innocent equation is the premise of this proof and while it seems initially unintuitive, on further inspection it makes logical sense. Where n is odd in this series the value of the sequence up to that point is 1. Conversely, where n is even the value is 0. Since ∞ ∉ 𝑂 ∪ the logical value for the sequence at the infinite point is the average of the two: 1 2⁄ . The next stage tackles another new sequence with the knowledge gained from the last. Here the sequence C 2 is used:
2
= 1 − 2 + 3 − 4 + 5 …
2 2 = (1 − 2 + 3 − 4 + 5 … ) + (1 − 2 + 3 − 4 + 5 … ) = 1 − 1 + 1 − 1 + 1
2 2
= 1
= 1
2⁄
1
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