[jasallen]

For those complaining about the verbosity of c#, here is an alternate c# representation:

    public static void Main()
    {
        const int n = 21;

        Observable.Generate<double,double>(1, x=>true, x=> 0.5 * (x + (n/x)), x=>x )
        .Scan(new {prv = 0d, cur = 0d}, (prev, curr) => new {prv = prev.cur, cur = curr})
        .FirstAsync(_ => Math.Abs(_.cur - _.prv) < 1E-10)
        .Select(_ => _.cur)
        .Subscribe(Console.WriteLine);

        // this is just to compare values, so is not part of the solution
        Console.WriteLine(Math.Sqrt(n)); 
        Console.ReadLine();
    }

That's is, everything you need to run the program (technically one line of code, line breaks are for readability). And its non-destructive, has no side effects and is pretty easy to read.

I would never argue that c# is a 'proper' functional language, but the poster above wrote in the style of idiomatic java, something a lot of c# devs do. I don't think c# has an 'idiomatic'. It is very flexible, and while never truly functional, can be also be used like a scala.