Is sum of Natural Numbers upto Infinity -1/12??

Mon, Feb 29, 2016

The Lighter Side

This is simple arithmetic progression. A Sum of natural numbers from 1 to n. The answer is n(n+1)/2. Atleast, this is what we were taught all throughout our schooling. So, if ‘n’ were to tend to infinity, summation should tend to infinity. Right? Wrong!! Yes, mathematicians are saying ‘no’. Is there some hidden mystery behind this?

Consider S = 1+2+3+4+5+6+7+8…..

Ok. If the answer is not infinity which may be by some invisible hand, atleast the sum should be positive. See ‘S’; all terms are positive. So the sum is positive. Right? Again a ‘no’.

How a double ‘no’???Are we being played by number theorists? Is this some pseudo-science? It can’t be negative. Never negative!! So let’s leave this sum which looks too tricky. Let me write something else. I always revered Ramanujan as one of the greatest mathematicians of all time. The sheer genius of Ramanujan in number theory always fascinated me. The great man often ignored proofs for many of the derivations. It’s like this; a genius looks at a problem, the path looks obvious, so he skips the steps, reaches the solution, but the mathematicians are still pondering on the left conjectures. So yes Ramanujan also did interesting mathematics in the field of infinite summation and the next statement shocks me!!! Ramanujan’s method for summation of numbers, points to the fact ‘S’= -1/12. Ramanujan? Did he not study basic formula n(n+1)/2? Or those divergent series stuff? But one more eminent mathematician’s work went into proving ‘S’=-1/12. This was “Riemann”. Yes he is well known for zeta functions and reputed as one of the best mathematicians of recent times; but -1/12??

With such great minds saying ‘S’=-1/12, I am now skeptical. Can the sum be -1/12? Can we try a method to prove this?

Let S = 1+2+3+4+5+6+7

Consider S1= 1-1+1-1+1-1+1-1…..

Now, this sum should be 0 or 1 based on number of natural numbers taken. If infinite numbers are even, S1=0, if odd S1=1. But, Riemann zeta function gives it a value of ½. Mathematical community too agrees that the sum is ½. How? At first instance, it feels like the whole community of mathematicians are playing a prank. It is like celebrating April fools day! But yes, serious mathematical work went into the proof. If you are interested to know, please go through Ramanujan’s summation principles and zeta function. Let me try a simple proof avoiding all the complexity.

S1=1-1+1-1+1-1+1…..

1-S1=1-(1-1+1-1+1-1+1…)

1-S1=1-1+1-1+1-1+1…..

1-S1=S1

So, S1=1/2

Several objections can be put like why not the alternative solution of 0 or 1? But as stated before, we need much powerful tools in mathematics like zeta functions to come to unique solution of ½. For now, we could agree S1=1/2.

Let S2=1-2+3-4+5-6+7…..

So, S2=1-2+3-4+5-6+7-8+9…..

S2=    1-2+3-4+5-6+7-8……. I have shifted RHS by a unit position

+ 2S2=1-1+1-1+1-1+1…..

Hence, 2S2=S1

Therefore, S2=1/4

Let’s come back to our sum of infinite numbers.

S=1+2+3+4+5+6+7+8+9…..

S2=1-2+3-4+5-6+7-8+9….

So, S-S2=4+8+12+16+20…..

Hence, S-S2=4(1+2+3+4+5+6+7+8….)

S-S2=4S

So, -S2=3S

And, S=-S2/3=-1/12

Amazing!!! Our sum is negative! It looks like god plays with numbers in a bizarre way. This shocking result is not known to many non-mathematicians. Number-theorists call it “One of the most remarkable formulae in science”. This summation is a secret of mathematics kept away from layman. Further, it is interesting to know ‘S’=-1/12 has been used to derive the equations in “string theory”, quantum field theory and in some complex analytics.

So now you know sum of positives can be negative. We knew much less during schooling about summation of numbers.  The result teaches how universe can be more complicated than we think and how we need to keep an open mind to learn more.

By: K Prakash Raju

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