Permalink Submitted by Jayanta Boral on May 11, 2017

Following the Infinite Series 1-1+1-1+...til infinity = 1/2 and 1+2+3+4+...till infinity = -1/12 and applying to a series 1+1+1+....till infinity, I get a different but interesting results. I am explaining it below -

Say S=1+1+1+1+.....
I can go on adding S vertically downwards also infinitesimally but with one right shift of 1 every time, so that we get the following -

S = 1+1+1+1+1+..... till Infinity
S = 0+1+1+1+1+1+.... till Infinity
S = 0+0+1+1+1+1+1+... till infinity
.
.
.
.
till Infinity

I added 0 just to show alignment otherwise the HTML was not able to arrange numericals as I wanted.

Now adding all above, we get

S+S+S+S+....Till Infinity = 1+2+3+4+5+....till Infinity = -1/12
or, S(1+1+1+1+... till Infinity) = -1/12
or, S*S=-1/12 or S^2=-1/12 Therefore, S= Sqrt(-1/12)

So, precisely what we get is sum of 1 infinite times is square root of a negative fraction. How do we explain this?

## Different Results

Following the Infinite Series 1-1+1-1+...til infinity = 1/2 and 1+2+3+4+...till infinity = -1/12 and applying to a series 1+1+1+....till infinity, I get a different but interesting results. I am explaining it below -

Say S=1+1+1+1+.....

I can go on adding S vertically downwards also infinitesimally but with one right shift of 1 every time, so that we get the following -

S = 1+1+1+1+1+..... till Infinity

S = 0+1+1+1+1+1+.... till Infinity

S = 0+0+1+1+1+1+1+... till infinity

.

.

.

.

till Infinity

I added 0 just to show alignment otherwise the HTML was not able to arrange numericals as I wanted.

Now adding all above, we get

S+S+S+S+....Till Infinity = 1+2+3+4+5+....till Infinity = -1/12

or, S(1+1+1+1+... till Infinity) = -1/12

or, S*S=-1/12 or S^2=-1/12 Therefore, S= Sqrt(-1/12)

So, precisely what we get is sum of 1 infinite times is square root of a negative fraction. How do we explain this?

Can anyone help?