Ramanujan First Problem

Srinivasa Ramanujan is one of my heroes. A brilliant mind from Southern India (Chennai) who had an incredible talent for numbers and unfortunately a very tragic and brief life. If you want to learn more about him I recommend reading The Man Who Knew Infinity instead of watching the movie. In the movie there is no, zero, mathematics. And if you learn about Ramanujan without appreciating his work in math it is as if you were learning about Einstein without appreciating his work in Physics.

In the book you'll find one of the very first problems proposed by Ramanujan when he was still a kid. It was an infinite series (95% of his work revolved around infinity) where he asked all the top mathematicians in the world if they could solve it. Simply stated, here it is:

What's the value of X? Top mathematicians all over the world jumped into this problem but with no success. They worked on it for months and months until they all, one by one, gave up. Finally Ramanujan gave away the solution. X = 3.

The mathematics behind the proof are pretty advanced. Only if the mathematicians 100+ years ago had access to a Surface Laptop with Visual Studio installed....

Writing a simple code to converge the series above is relatively easy. The equation is very suitable for a recursive solution, and hopefully the convergence will happen quick enough that the stack won't blow up. In my 2-lines recursive function I use a max seed to stop the recursion (remember, this is an infinite sum). Running the code for, say, 70 iterations quickly shows where it is converging towards. Code is below, output down below, cheers, Marcelo.

class Program
{
static void Main(string[] args)
{
int max = 70;
for (int i = 0; i <= max; i++)
{
Console.WriteLine("For terms = {0}, RFP = {1}", i, RamanujanFirstProblem(2, i));
}
}
static double RamanujanFirstProblem(int currentSeed, int maxSeed)
{
if (currentSeed >= maxSeed) return 0.0;
return Math.Sqrt(1 + currentSeed * RamanujanFirstProblem(currentSeed + 1, maxSeed));
}
}

Output:

For terms = 0, RFP = 0

For terms = 1, RFP = 0

For terms = 2, RFP = 0

For terms = 3, RFP = 1

For terms = 4, RFP = 1.73205080756888

For terms = 5, RFP = 2.23606797749979

For terms = 6, RFP = 2.55983016530012

For terms = 7, RFP = 2.7550532613299

For terms = 8, RFP = 2.86710292823774

For terms = 9, RFP = 2.92917335973214

For terms = 10, RFP = 2.96272300427958

For terms = 11, RFP = 2.98055375021182

For terms = 12, RFP = 2.98992036063105

For terms = 13, RFP = 2.99480026926621

For terms = 14, RFP = 2.9973274414787

For terms = 15, RFP = 2.99863031788908

For terms = 16, RFP = 2.99929967806766

For terms = 17, RFP = 2.99964261492875

For terms = 18, RFP = 2.99981791758458

For terms = 19, RFP = 2.99990736080556

For terms = 20, RFP = 2.99995292420987

For terms = 21, RFP = 2.99997610308338

For terms = 22, RFP = 2.99998788059949

For terms = 23, RFP = 2.99999385870344

For terms = 24, RFP = 2.9999968903158

For terms = 25, RFP = 2.99999842644538

For terms = 26, RFP = 2.99999920423618

For terms = 27, RFP = 2.99999959779522

For terms = 28, RFP = 2.99999979681498

For terms = 29, RFP = 2.99999989740273

For terms = 30, RFP = 2.999999948216

For terms = 31, RFP = 2.99999997387325

For terms = 32, RFP = 2.99999998682296

For terms = 33, RFP = 2.99999999335639

For terms = 34, RFP = 2.99999999665146

For terms = 35, RFP = 2.99999999831275

For terms = 36, RFP = 2.99999999915007

For terms = 37, RFP = 2.99999999957197

For terms = 38, RFP = 2.99999999978449

For terms = 39, RFP = 2.99999999989152

For terms = 40, RFP = 2.9999999999454

For terms = 41, RFP = 2.99999999997253

For terms = 42, RFP = 2.99999999998618

For terms = 43, RFP = 2.99999999999305

For terms = 44, RFP = 2.9999999999965

For terms = 45, RFP = 2.99999999999824

For terms = 46, RFP = 2.99999999999912

For terms = 47, RFP = 2.99999999999956

For terms = 48, RFP = 2.99999999999978

For terms = 49, RFP = 2.99999999999989

For terms = 50, RFP = 2.99999999999994

For terms = 51, RFP = 2.99999999999997

For terms = 52, RFP = 2.99999999999999

For terms = 53, RFP = 2.99999999999999

For terms = 54, RFP = 3

For terms = 55, RFP = 3

For terms = 56, RFP = 3

For terms = 57, RFP = 3

For terms = 58, RFP = 3

For terms = 59, RFP = 3

For terms = 60, RFP = 3

For terms = 61, RFP = 3

For terms = 62, RFP = 3

For terms = 63, RFP = 3

For terms = 64, RFP = 3

For terms = 65, RFP = 3

For terms = 66, RFP = 3

For terms = 67, RFP = 3

For terms = 68, RFP = 3

For terms = 69, RFP = 3

For terms = 70, RFP = 3