Mathemagics 1: Challenging Ramanujan's Summation

 

What is 1+2+3+…..+9+10=?  Well, it’s quite easy 55. In fact any such sum of finite natural numbers can solved by the formula S=  (n(n+1))/2 

But if I ask you what is the sum of this same series infinitely?

What is 1+2+3+……∞ = ?

You will be shocked to know that this series equals to  -1/12

What the frac! The sum of infinite positive numbers is a negative number? And that also a fraction. Well let’s see how this is possible:

Also Read: Mathemagics 1: Interesting facts about Maths

To prove this statement we have to consider these two series:

·      ·     A= 1-1+1-1+1…..

·      ·   B= 1-2+3-4+5…..

Now subtracting A from 1, we get

Ø  1-A = 1-(1-1+1-1+1….)

Ø  1-A = 1-1+1-1+1…

Well if you see, this is exactly the same series that we started with

Ø  1-A = A

Ø  2A = 1

Ø A = 1/2

Next we subtract B from A and we get

Ø  A-B = (1–1+1–1+1–1⋯) — (1–2+3–4+5–6⋯)

Ø  A-B = (1–1+1–1+1–1⋯) — 1+2–3+4–5+6⋯

Ø  A-B = (1–1) + (–1+2) +(1–3) + (–1+4) + (1–5) + (–1+6)

Ø  A-B = 0+1–2+3–4+5

So the result is B

Ø      A-B = B

Ø      A = 2B

Ø      B = A/2

Ø ·   B = 1/4

Now we take another series which is the question itself:

·        ·   C = 1+2+3+4+5+6⋯

 

Now subtract C from B and we get

Ø  B-C = (1–2+3–4+5–6⋯)-(1+2+3+4+5+6⋯)

Ø  B-C = (1-2+3-4+5-6⋯)-1-2-3-4-5-6⋯

Ø  B-C = (1-1) + (-2-2) + (3-3) + (-4-4) + (5-5) + (-6-6) ⋯

Ø  B-C = 0-4+0-8+0-12⋯

Ø  B-C = -4-8-12⋯

Ø  B-C = -4(1+2+3)⋯

Ø  B-C = -4C

Ø  B = -3C

 

And now you are going to see the thing….

Ø  1/4 = -3C

Ø  1/-12 = C 

Ø  C = -1/12

So Ramanujan’s summation conclude that

·       1+2+3+4+5….+∞ = -1/12

 

So this is what mathematician Ramanujan has derived.

Now I shall derive a different value of the series in a similar way and here it is

For this I shall take another series

·        D = -1-2-3-4-5…..

Next subtract D from  A and see the magic

Ø  A-D = (1–1+1–1+1–1⋯) - (-1-2-3-4-5–6⋯)

Ø  A-D = (1–1+1–1+1–1⋯) + 1+2+3+4+5….

Ø  A-D = (1+1) + (-1+2) + (1+3) + (-1+4) + (1+5) + (-1+6)+….

Ø  A-D = 2+1+4+3+6+5….

Ø  A-D = 1+2+3+4+5+6….

Hence we can say that

Ø       A-D = C

Ø      D = A-C

Ø   D =  (1/2)-(-1/12)   = 7/12

Now taking -1 common from the series D, we get

Ø     D = - (1+2+3+4+5…) = -C = 7/12

Ø  C = -7/12

So this is how we got a different value of the Ramanujan series. Isn’t that interesting? 

But then comes the question – is it even true? How can the sum of infinite positive numbers can be a negative fraction in the world Well, Ramanujan Summation is a technique invented by the great Indian mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. This summation is not a sum in traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is not defined – Wikipedia.

To know more about Srinivasa Ramanujan the great mathematician you can check out these two famous books on Ramanujan:

The man who knew the infinity

Mathematics Wizard