The proof is rather simple. Before we get to that, let us try and understand a couple of other series first -
Consider the following infinite summation (X):
X = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 โฆ
Rearranging the above equation, we get:
X = 1 - (1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 โฆ)
If you look at the term inside the brackets, you will realize it is our original series โXโ. So letโs substitute that:
X = 1 - X
2X = 1
X = ยฝ
Now letโs consider another summation (Y) :
Y = 1 - 2 + 3 - 4 + 5 +6 โฆย -A
Writing it in another way by adding 0 to both sides, we get:
0 + Y = 0 + 1 - 2 + 3 - 4 + 5 + 6โฆ
or simply,
Y = 0 + 1 - 2 + 3 - 4 + 5 + 6โฆย -B
Adding the two equations A + B:
Y + Y = (1 - 2 + 3 - 4 + 5 โฆ) + (0 + 1 - 2 + 3 - 4 + 5 โฆ)
Grouping the corresponding terms within the brackets, we get:
2Y = 1 + 0 - 2 + 1 + 3 - 2 - 4 + 3 + 5 - 4 โฆ
2Y = 1 - (2-1) + (3-2) - (4-3) + โฆ
2Y = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1โฆ
But the summation on the right hand side is X as we previously saw, so letโs substitute it:
2Y = X
2Y = ยฝ
Y = ยผ
Finally, letโs consider our original series i.e. the sum of all natural numbers:
S = 1 + 2 + 3 + 4 + โฆ
We defined Y earlier as:
`Y = 1 - 2 + 3 - 4 + โฆ
Subtracting Y from S:
S - Y = 1 - 1 + 2 + 2 + 3 - 3 + 4 + 4 + โฆ
S - Y = 4 + 8 + 12 + 16 + โฆ
We just calculated the value of Y which is equal to ยผ. So letโs substitute it:
S - ยผ = 4 x (1 + 2 + 3 + 4 + โฆ)
S - ยผ = 4S
3S = -ยผ
S = -1/12
So there we go! We now have the proof.
Looks like a clever maneuver. Is this a mathematical trick?
Answer isย No.
It actually appears in many areas of physics.
The only thing to notice here is - we are applying the rules of regular algebra to a divergent infinite series.
A convergent seriesย is the one in which the sum keeps converging to a particular definite number as you keep adding more numbers to it.ย Divergent seriesย is the opposite of that. We will discuss more about it in the next part.