Numerology - Infinite Sum of Integers

Infinite Sum of Integers

Many people will be surprised to learn that the sum of all integers ie 1 + 2 + 3 + 4 ..... to infinity actually equates to -1/12. To prove this we begin by looking at the following two series.

First Series

Let us define S₀ as the following infinite series:

S₀ = 1 - 1 + 1 - 1 + 1 - 1 .......... ∞

This sum will either be 1 or 0 depending on the number of elements, however we are dealing with an infinite series so the term "number of elements" becomes meaningless. However if we do the following:

1 - S₀ = 1 - 1 + 1 - 1 + 1 - 1 .......... ∞

Hence

1 - S₀ = S₀

Therefore

S₀ = 1/2

Second Series

New let us define S₁ as the following infinite series:

S₁ = 1 - 2 + 3 - 4 + 5 - 6 .......... ∞

However if we add two series together by moving the second over 1 digit we get:

S₁ = 1 - 2 + 3 - 4 + 5 - 6 .......... ∞

Thus

2S₁ = 1 - 1 + 1 - 1 + 1 ......... = 1/2

Hence

S₁ = 1/4

Sum of all Integers

Now let us look at the sum of all integers:

S_all = 1 + 2 + 3 + 4 + 5 + 6 .......... ∞

If we now deduct S₁ we get

S_all - S₁ = (1 - 1) + (2 + 2) + (3 - 3) + (4 + 4) + (5 - 5) + (6 + 6) .......... ∞

Hence

S_all - S₁ = 4 + 8 + 12 .......... ∞ = 4S_all

Which leads to the conclusion that

S_all = -1/12

Updated: 10/12/2023