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 series**ย is the one in which the sum keeps converging to a particular definite number as you keep adding more numbers to it.ย **Divergent series**ย is the opposite of that. We will discuss more about it in the next part.*