### Ramanujan's Infinity Series

## Simple proof of Ramanujan's Infinite Series

**Srinivasa Ramanujan**was an

**He was mostly famous for his work in the fields of**

*Indian mathematician.**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

So,

2 S2 = 1/2

Now, lets' take the infinite series,

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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