Nested radicals?

The first number: x = {1+[1+(1 ... )^(1/2)]^(1/2)}^(1/2) satisfies x²=1+x which, given x>0, allows only x=(1+√5)/2, the golden ratio. Of course one has to show convergence, but that is not too hard, or you can appeal to Herschfeld’s convergence theorem (look it up).

The second number {1+[2+(3 ... )^(1/2)]^(1/2)}^(1/2) is more interesting. It is commonly known as the Nested Radical Constant, although Herschfeld called it the Kasner number. A spreadsheet or calculator quickly gives a good approximation. I get about 3 digits for every 4 terms. However, according to the Online Encyclopedia of Integer Sequences http://oeis.org/A072449 "No closed-form expression is known for this constant". I haven't been able to improve on that but if anyone here can, please speak up and be famous!