Ramanujan, The greatest ever Indian mathematician.

Only recently I read a biography “The Man Who Knew Infinity”, on Ramanujan, the greatest ever Indian mathematician, . It was amazing and poignant at the same time. Some of the problems mentioned in the book, revved up my aging mind. I tried to prove one of his equations. I am sure this being one of the simpler equations of Ramanujan, many would have solved it during the last century itself. Still I feel elated that I was able to share a very small part of his genious. Ofcourse, unlike Ramanujan, I had a formal education upto master’s level in engineering in one of the IITs. If only we had IITs in those days ……..? I am attaching two proofs for Ramanujan’s equation, one of which I call as Derivation. I feel very happy to share the same with my readers. Happy to be associated with great minds of India.
L V Nagarajan

Solution for the Problem by Sri Ramanujan

1. To Prove

(x + n + a) = √[ax +(n+a)**2 +x√[a(x+n) +(n+a)**2 + (x+n)√[a(x+2n) +(n+a)**2 + (x+2n) √ etc ….

Proof
Let Ax = √[ax +(n+a)2 +x√[a(x+n) +(n+a)2 + (x+n)√[a(x+2n) +(n+a)2 + (x+2n) √etc ….
Then, We may write
Ax = √[ax +(n+a)**2 +x A x+ n]
i.e. Ax**2 = [ax +(n+a)**2 +x A x+ n]
i.e. Ax**2 - (n+a)**2 = [ax + x A x+ n] = [ a + A x+ n ] x
i.e. [Ax - (n+a)] . [Ax + (n+a)] = [ a + A x+ n ] x
i.e. [Ax - (n+a)] / x = [ a + Ax+ n ] / [Ax + (n+a)] = k (say) —– (1)

Hence, Ax = kx + n + a; And so, A x+n = k(x + n) + n + a ———(2)

Substituting in the second part of eqn(1) above,
[a + k(x + n) + n + a] / [kx +n +a + n+ a] = k
i.e x k**2 + (2n + 2a – x – n) k – (n + 2a) = 0
i.e. x k**2 + (n + 2a – x) k – (n + 2a) = 0
i.e. (xk + n + 2a) (k – 1) = 0, giving the values, k =1 or kx = - n – 2a
Substituting in (2)
Ax = x + n + a , or Ax = - a, (not admissible)
Hence proved


2. The above is a proof. Let us call the following as a derivation.

Derivation
(x +n+ a) = √ [ (x + n + a)**2 ]
= √ [ x**2 + ( n + a)**2 + 2x(n + a) ]
= √ [ ax + x**2 + ( n + a)**2 + x(2n + a) ]
= √ [ ax + ( n + a)**2 + x(x + n +n + a) ] ——–(i)

Following Eqn (i) above, we may write:
(x+n +n + a) = √ [ a(x+n) + ( n + a)**2 + (x+n)(x + 2n + n + a) ] ——(ii)

From Eqns (i) and (ii), we can recursively write:
(x +n +a) = √[ax + ( n + a)**2 + x√[a(x+n) + ( n + a)**2 + (x+n)√[a(x + 2n) + (n+a)**2 + (x+2n)√…

Hence Derived

3. To Find the value of √[1+2√[1+3√[1+4√[1+5√[1+ …….

(n+1)**2 = n**2 + 2n + 1
= 1 + n(n+2)
i.e.
n + 1 = √ [1 + n(n+2]]
Hence we may write,
3 = √ [1 + (2 x 4)] and
4 = √ [1 + (3 x 5)] and
5= √ [1 + (4 x 6)], etc

Hence,
3= √[1+2√[1+3√[1+4√[1+5√[1+ …….



L V Nagarajan
01 July 2008

Once There Were Rivers

Once there were rivers.
L V Nagarajan

Rivers are the worst sufferers of human greed. I read somewhere that about 80% of the rivers of the world do not reach the sea at all. They become dry miles before they reach their natural end. River beds are used as illegal sand quarries and later as illegal real estates. What do we do about it? There should be an international law to limit the utilization of river water to, say, 95% and a minimum of 5% of water, as a rule should be discharged into the sea. A norm should be developed for graded utilization of river water all along its route upto the sea, irrespective of political boundaries it passes through. Let us think of our ancient cultures which worshiped rivers, especially its source and its point of collusion with the sea. Let us not pollute the rivers and let us keep their banks and the beds clean and clear. Let us preserve our rivers for our future generations.

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