2 = 1 4⁄ This sequence also causes unease with a strange blend of positive and negative integers culminating in a fraction. Fortunately, this brings us to a position from which we can prove our titular summation:
= 1 + 2 + 3 + 4 + 5 …
− 2
= (1 + 2 + 3 + 4 + 5 … ) − (1 − 2 + 3 − 4 + 5 … ) = 4 + 8 + 12 +
16 + 20 …
− 2
= 4 = − 1
4⁄
3 = −1
4⁄
∴ = −1
12⁄
And there we have it: a concise and intuitive proof of what is a deeply unintuitive sum. For all that work, however, something feels wrong: incomplete. This foundation-rocking piece of mathematics was all so simple! Surely there must be more to it that that; I, like that part of you deep inside, crave intelligible squiggles littering the page. Fear not, however, for I mentioned Euler’s proof: a forest of differentiation, sigmas and Riemann-Zeta functions. Despite this, the mathematics itself is quite rational and very satisfying.
Euler’s Proof:
Euler’s proof begins with a seemingly unrelated series and the proof of its constant, K :
= 1 + + 2 + 3 …
+ 1 = 1 + + 2 + 3 … =
− = 1 = (1 − )
∴ = 1 + + 2 + 3 … = 1
; < 1
(1 − ) ⁄
Following this, we differentiate K:
𝑥 = 1 + 2 + 3 2 + 4 3 … = 1
(1 − ) 2 ⁄
And let = −1 :
1 − 2 + 3 − 4 + 5 … = 1
2 2 ⁄ = 1
4⁄ = 2
2
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