1 + 2 + 3 + ... = -1/12
What a normal person would think of this sum 1 + 2 + 3 + 4....... ? Everyone will agree that the sum is increasing and increasing all the way to infinity. That means that there is no way we can assign a finite value but that's not how Ramanujan saw it, In his paper to G.Hardy he wrote this theorem 1 + 2 + 3 + 4.... = -1/12, though it was earlier proved by Euler and Riemann, Ramanujan did it separately with no formal training in mathematics.
This result is used in may areas of physics like string theory.
Here is how it is proved in simple way.
Let us define some series of sum we will call as X and Y as follows :
X = 1 - 1 + 1 - 1 + 1 - 1 + .... (1)
Y = 1 - 2 + 3 - 4 + 5 - 6 + .... (2)
Now let, S = 1 + 2 + 3 + 4 + 5 + .... (3)
Analysis of equation (1)
If we analyse equation (1) we observe that if the numbers in the series are even then we get X = 0, but in the case of odd number it is X = 1, so what will be the sum all the way up to infinity, we don't know so we take the average of the two,
Hence we can say that X = 1/2, there are other ways to prove this but for the time being consider the intuition.
X = 1/2 (4)
Analysis of equation (2)
Now, the trick I am going to use is that I will add Y to itself but by shifting one place:
Y = 1 - 2 + 3 - 4 + 5 - 6 + ....
Y = 1 - 2 + 3 - 4 + 5 + ....
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2Y = 1 - 1 + 1 - 1 + 1 - 1 + ....
Now the R.H.S = X = 1/2 ---- using (4)
Therefore, Y = 1/4 (5)
Now we have all the pre-requirements to prove the result;
Subtracting equation (2) from equation (3);
S = 1 + 2 + 3 + 4 + 5 + ....
Y = 1 - 2 + 3 - 4 + 5 + ....
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S - Y = 4 + 8 + 12 + .....
S - Y = 4 ( 1 + 2 + 3 + ....
S - Y = 4S ---- using (3)
-3S = Y ---- Subtracting 4S and adding Y on both sides;
S = -(Y)/3 ---- Dividing both sides by -3.
Therefor by substituting the value of Y from equation (5) we get the value of S
S = -1/12;
Hence proved.