Summation of all natural numbers is (-1/12), How? Let’s see.

Unbelievable isn’t it ?

How can all the positive numbers can add up to give a negative number?

Let us assume

P = 1-1+1-1+1-1+….

Q = 1-2+3-4+5-6+…

R = 1+2+3+4+5+6+…

All the series are infinite non ending series.

Now, we will try to evaluate P. What could be the possible answers to P? 1 or 0 ? But what if I tell you that answer to P is actually average of both the values

P = 1-1+1-1+1-1+… 1-P = 1-[1-1+1-1+1-1+1-1+…]

1-P = 1-1+1-1+1-1+…

So we can say that,

P = 1-P Which implies that

2*P = 1

P= 1/2 (More discussion on this result at the end of this article)

Now, Let’s go back to our original problem (Value of S?)

2*Q= (Q + Q )or we can say 2*Q= Q + (0+Q) and we can add them vertically column by column as shown below and the result we get is P

2*Q= 1-2+3-4+5-6+…..

+0+1-2+3-4+5-6+…..

2*Q= 1-1+1-1+1-1+1-1+1-1+…= P

So, 2*Q=P and P=1/2 so we get the value of Q

Q = 1/4

Now, R-Q done by same method vertically column by column without introducing Zero like we did last time.

R-Q= 1+2+3+4+5+6+…

-[1-2+3-4+5-6+7-…]

R-Q=0+4+0+8+0+12+0+16+……

R-Q = 4 *(1+2+3+4+5+6+….)

R-Q = 4 * (R)

(-3)* R = Q

and as we already found Q = 1/4

So, (-3)*R= 1/4

R= (-1/12)

Hence proven 🙂

Now, Let’s talk about series P

I find it very fascinating that sum of 1 and -1 is actually 1/2, and if we can visualize this somehow then I think it will be easier to take that 1 and -1 could add up to be 1/2 if added infinite time.

Assume that you are in a room and on command of 1 you switch on the light and on -1 you switch of the light. Initially frequency is once per second and slowly increase the frequency and somewhere in between you’ll reach to point where that room will feel like disco where white light flickers, if that helps in your assumption assume that hapening even faster.

At this point you won’t be able to see “darkness” neither you’ll see full intensity “whiteness”. What you’ll be able to see if actually average of both that is darkness(-1)and full intensity light(1).You’ll be seeing their summation and intensity of that light would be 1/2 of highest intensity.

One more example on this, whenever you point camera at your TV or laptop screens you see different bands travelling up and down on your screen. you can assume darker band -1 and intense band as 1. With your naked eye you are just seeing summation of both the bands.

This physical application or interpretation of series P always leaves me thinking where else this could be in mother nature? where else?

Bibliography

God speaking