Ramanujan's Infinity Series

                           Simple proof of Ramanujan's Infinite Series


Srinivasa Ramanujan was an Indian mathematician. He was mostly famous for his work in the fields of number theory and algebaric geometry. He gave the popular infinite series summation. Here we will prove his infinite series according to Ramanujan's theory.
He proved that,
                                           1+2+3+4+5+6+.........+∞ = -1/12 

It is called the Ramanujan's Infinite series.

PROOF :

Lets' consider a series, like,

S1 = 1-1+1-1+1-1+1-1+1.......+∞ = 0  ( If the number of elements in the series are even)
                                                = 1  ( If the number of elements in the series are odd)

Now, here we can't a fixed value of this series. We are getting two different values in case of two different cases.
So, lets take the average value of the series.

Average value of S1 = (0+1)/2 = 1/2

Lets' take one more series, like,

S2 = 1-2+3-4+5-6+7-8........+∞ 

Now, sum up this series S2 with itself (S2), in this manner

S2 = 1-2+3-4+5-6+7-8.......+
           1-2+3-4+5-6+7-8......+
         1-1+1-1+1-1+1-1.........+
Here we just shift the second row in one element right side and get the first series S1, and we know the value of S1, which is 1/2

So, S2+S2 = S1 =1/2

       2 S2 = 1/2
          S2 = 1/4

Now, lets' take the infinite series,

S3 = 1+2+3+4+5+6+.......+

Now, Lets' do, 

  S3 - S2 =   (1+2+3+4+5+6+......+∞) - (1-2+3-4+5-6.......+∞)

Doing element wise subtraction, we can get the series,

S3 - S2 = 0+4+0+8+0+12+0+16+0+20+0+......+
S3 - S2 = 4+8+12+16+20+.....+
S3 - S2 = 4 (1+2+3+4+5+......+∞)
S3 - S2 = 4 S3
S3 - 1/4 = 4 S3    ( S2 = 1/4)
S3 - 4 S3 = 1/4
-3 S3 = 1/4
S3 = -1/12

Hence, it is proved that,

                                       S3 = 1+2+3+4+5+6+.......+ = -1/12 
                  

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