#### 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 S_{0}as the following infinite series:S

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:
_{0}= 1 - 1 + 1 - 1 + 1 - 1 .......... ∞1 - S

Hence
_{0}= 1 - 1 + 1 - 1 + 1 - 1 .......... ∞1 - S

Therefore
_{0}= S_{0}S

_{0}= 1/2Second Series

New let us define S_{1}as the following infinite series:S

However if we add two series together by moving the second over 1 digit we get:
_{1}= 1 - 2 + 3 - 4 + 5 - 6 .......... ∞S

_{1}= 1 - 2 + 3 - 4 + 5 - 6 .......... ∞S

Thus
_{1}= 1 - 2 + 3 - 4 + 5 - 6 .......... ∞2S

Hence
_{1}= 1 - 1 + 1 - 1 + 1 ......... = 1/2S

_{1}= 1/4Sum of all Integers

Now let us look at the sum of all integers:
S

If we now deduct S_{all}= 1 + 2 + 3 + 4 + 5 + 6 .......... ∞_{1}we getS

Hence
_{all}- S_{1}= (1 - 1) + (2 + 2) + (3 - 3) + (4 + 4) + (5 - 5) + (6 + 6) .......... ∞S

Which leads to the conclusion that
_{all}- S_{1}= 4 + 8 + 12 .......... ∞ = 4S_{all}S

_{all}= -1/12Updated: 10/12/2023