The Ramanujan Summation…

Srinivasa Aiyangar Ramanujan

Ramanujan summation – as you can read from Wikipedia – is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.

According to Ramanujan, if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12. Yup, -0.08333333333.

\( \xi \left( { – 1} \right) = 1 + 2 + 3 + 4 + \ldots = – \frac{1}{{12}}\)

Don’t believe me? Keep reading to find out how I prove this, by proving two equally crazy claims:

\(1 – 1 + 1 – 1 + 1 – 1 + \ldots = 1/2 \) \(1 – 2 + 3 – 4 + 5 – 6 + \ldots = 1/4\)

First off, the bread and butter. This is where the real magic happens, in fact the other two proofs aren’t possible without this.

I start with a series, A, which is equal to 1–1+1–1+1–1 repeated an infinite number of times. I’ll write it as such:

\(A = 1 – 1 + 1 – 1 + 1 – 1 + \ldots \)

We can write, now:

\(1 – A = 1 – \left[ {1 – 1 + 1 – 1 + 1 – 1 + \ldots } \right]\)

so, we have:

\(1-A=A\)

and then:

\(A=\frac{1}{2}.\)

This little beauty is Grandi’s series, called such after the Italian mathematician, philosopher, and priest Guido Grandi.

Now, let to prove that:

\(1 – 2 + 3 – 4 + 5 – 6 + \ldots = 1/4\)

We start the same way as above, letting the series B =1–2+3–4+5–6⋯. Then we can start to play around with it. This time, instead of subtracting B from 1, we are going to subtract it from A. Mathematically, we get this:

\(A – B = \left( {1 – 1 + 1 – 1 + 1 – 1 + \ldots } \right) – \left( {1 – 2 + 3 – 4 + 5 – 6 + \ldots } \right)\)

and then:

\(A – B = \left( {1 – 1 + 1 – 1 + 1 – 1 + \ldots } \right) – 1 + 2 – 3 + 4 – 5 + 6 – \ldots \)

Then we shuffle the terms around a little bit, and we see another interesting pattern emerge.

\(A – B = \left( {1 – 1} \right) + \left( { – 1 + 2} \right) + \left( {1 – 3} \right) + \left( { – 1 + 4} \right) + \left( {1 – 5} \right) + \left( { – 1 + 6} \right) + \ldots \) \(A – B = 0 + 1 – 2 + 3 – 4 + 5 + \ldots \)

Once again, we get the series we started off with, and from before, we know that A = 1/2, so we use some more basic algebra and prove our second mind blowing fact of today.

\(A – B = B \Leftrightarrow A = 2B \Rightarrow 1/2 = 2B \Leftrightarrow B = 1/4\)

And voila! This equation does not have a fancy name, since it has proven by many mathematicians over the years while simultaneously being labeled a paradoxical equation. Nevertheless, it sparked a debate amongst academics at the time, and even helped extend Euler’s research in the Basel Problem and lead towards important mathematical functions like the Riemann Zeta function.

Now for the icing on the cake, the one you’ve been waiting for, the big cheese. Once again we start by letting the series C = 1+2+3+4+5+6⋯, and you may have been able to guess it, we are going to subtract C from B.

\(B – C = \left( {1 – 2 + 3 – 4 + 5 – 6 + \ldots } \right) – 1 – 2 – 3 – 4 – 5 – 6 – \ldots \)

Because math is still awesome, we are going to rearrange the order of some of the numbers in here so we get something that looks familiar, but probably wont be what you are suspecting.

\(B – C = \left( {1 – 1} \right) + \left( { – 2 – 2} \right) + \left( {3 – 3} \right) + \left( { – 4 – 4} \right) + \ldots \)

and then,

\(B – C = 0 – 4 + 0 – 8 + 0 – 12 + \ldots = – 4 – 8 – 12 – \ldots \)

At this point, we can write:

\(B-C=-4(1+2+3+\ldots)\)

that is to say

\(B-C=-4C\)

and finally

\(B=-3C\)

And since we have a value for B=1/4, we simply put that value in and we get our magical result:

\(C=-1/12\)

or, in other words:

\( \xi \left( { – 1} \right) = 1 + 2 + 3 + 4 + \ldots = – \frac{1}{{12}}\)

Now, why this is important… Well for starters, this result is used in string theory. More. The Ramanujan Summation also has had a big impact in the area of general physics – specifically in the solution to the phenomenon know as the Casimir Effect. Hendrik Casimir predicted that given two uncharged conductive plates placed in a vacuum, there exists an attractive force between these plates due to the presence of virtual particles bread by quantum fluctuations. In Casimir’s solution, he uses the very sum we just proved to model the amount of energy between the plates. And there is the reason why this value is so important.