Ramanujan Summation

An overated mistake ?

In 1918 ,In a paper submitted by Srinivasa Ramanujan there was some highly controversial summation which socks the world of mathematics at that point of time.

but,
we know that sum of “finite” consecutive natural numbers is n(n+1)/2 ,And when n →∞
it obviously should be infinity ,
but,
the answer as given by Ramanujan, and later verified by another notable Mathematician Reimann, is – 1/12.Yes, not only is the answer a real number, it is a "negative” rational number!

So, were Ramanujan and Reimann wrong!

Let us consider few series,

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 …….. = 1/4

C = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 ………..

C = 0 + 1 - 2 + 3 - 4 + 5 - 6 + 7 …………

(Now add the two series in the certain way that 1 and 0, -2 and 1 , 3 and -2 , -4 and 3 and so on....)

2 C = 1 - 1 + 1 - 1 + 1 - 1 + 1 ………

-2 C = - 1 + 1 - 1 + 1 - 1 + 1 - 1 ……. (Multiplying -1 on both side )

1 - 2 C= 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 …… (Adding +1 on both side )

(Here the pattern re-appears)

1 - 2 C= 2C

1 = 4 C

C = 1/4

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 …….. = 1/4

Using this result we obtain ,

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

let, 

S= 1 + 2 + 3 + 4 + 5 + 6 + ..... _____________(1)

4S=     4       + 8    + 12 + ....._____________(2)

Now substracting (2) from (1) we get
(substract in a certain way that 2 and 4,4 and 8, 6 and 12 and so on.... )

S - 4S = 1 - 2 + 3 - 4 + 5 - 6 +.....

-3S = 1 /4    (from above series)

S= -1/12

Boom! 

Now, there are many steps may seem rather doubtful, but after multiple observations, we can find that they do make “some” sense.

Infinity is an extremely confusing Mathematical term, and when it is associated with a series, it does not generally give us the answer we expect. Such is the legacy of this term. Until someone actually comes up with a proper contradiction, we won’t take Ramanujan Summation as an overrated mistake, but rather as an astounding step in the ever-intriguing world of Mathematics!

-:THANKS:-

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