sum of infinite roots? What is the sum of this infinite series of roots:
$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4 + \cdots+\sqrt{\infty}}}}}$$

This is an interesting expression because the increase created by the addition of the next term is offset by the square root introduced with the next term. the key is determining what the relationship between these two is exactly

I know it converges on an irrational number around 1.75 as I have tried computing it for a few small iterations. But I was wondering how to algebraically simplify this expression.

I have tried many things and ways to manipulate the expression but to no avail.
This is the Nested Radical Constant
, which does converge, but whose closed-form expression is still unknown. See also Somos's Quadratic Recurrence Constant
 for a similar constant, involving their product.
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Hint:
$$
a_{1} = \root{1 + \root{a_{2}}}\,,\quad
a_{2} = \root{2 + \root{a_{3}}}\,, \ldots\,,
a_{n} = \root{n + \root{a_{n + 1}}}
$$
Then, $\ds{a_{n + 1} = \pars{a_{n}^{2} - n}}^{2}$. $\tt\mbox{Calculate}$ $a_{1}$.