<s>[INST] Sum of consecutive square roots inside a square root $$\large\sqrt{1+\sqrt{1+2+\sqrt{1+2+3+\sqrt{1+2+3+4+\cdots}}}}$$
I saw this somewhere in the internet but, the website didn't provide me any further information. What is the sum of the equation above? What is it called?
 [/INST] 
What is the sum of the equation above? What is it called?

It's neither an equation nor a sum, but is a continued radical defined as the limit   (as $n\to\infty$) of the sequence of the nested radicals
$$
u_n = \sqrt{a_1+\sqrt{a_2 + \cdots + \sqrt{a_n}}} 
$$
where $a_n = \sum_{i=1}^n i = n(n+1)/2$ (the $n^{th}$ triangle number), $n = 1,2,3,...$  

That this limit exists follows from a theorem proved by T. Vijayaraghavan (1927):


For any sequence of nonnegative reals $(a_n)_{n=1,2,3,...}$, the sequence of nested radicals $(u_n)_{n = 1,2,3,...}$ with
  $$u_n = \sqrt{a_1+\sqrt{a_2 + \cdots + \sqrt{a_n}}}$$
   converges if and only if there exists a finite upper limit 
  $$\overline{\lim}\ {\left(\frac{\log{a_n}}{2^n}\right)} < \infty.$$

This clearly holds in the present case, because
$$\overline{\lim}\ {\left(\frac{\log{\frac{n(n+1)}{2}}}{2^n}\right)}  = 0.
$$

To  bound the error of $u_n$ as an approximation of the limit, the following consequence of a theorem proved by Herschfeld (1935) can be used:

If $a_i \gt 0 \ \ (i = 1,2,3,...)$, then for all $n \ge 1$,
  $$0 \le u_{n+1}-u_n \le \frac{1}{2^n}\sqrt{\frac{a_{n+1}}{a_1 a_2 \cdots a_n}}. $$

Substituting $a_i = i(i+1)/2$ and simplifying then gives
$$0 \le u_{n+1}-u_n \le \frac{1}{n!}\sqrt{\frac{n+2}{2^{n+1}}} ,
$$
whence, noting that $n! \ge \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$,
$$0 \le u_{n+1}-u_n \le A \left(\frac{B}{n}\right)^n
$$
where $A = \sqrt{\frac{3}{4\pi}} = 0.4886...$ and $B = \frac{e}{\sqrt{2}} = 1.922...$
Now for all $m > n > 1$,
$$\begin{align}
0 \le u_m - u_n & = (u_m - u_{m-1}) + (u_{m-1} - u_{m-2}) + \cdots + (u_{n+1} - u_n) \\
& \le A \left (  \left (\frac{B}{m-1}\right )^{m-1} + \left (\frac{B}{m-2}\right )^{m-2} + \cdots + \left (\frac{B}{n}\right )^{n}     \right )\\
& \le A \left (  \left (\frac{B}{n}\right )^{m-1} + \left (\frac{B}{n}\right )^{m-2} + \cdots + \left (\frac{B}{n}\right )^{n}     \right )\\
& \le A \left (\frac{B}{n}\right )^n \frac{1 - (\frac{B}{n})^{m-n}}{1 - \frac{B}{n}}\tag{*}
\end{align}
$$
where the geometric progression was summed in the last step (noting that $B/n < 1$ for all $n > 1$).
Finally, taking limits with $m \to \infty$ and letting $u = \lim_{m \to\infty}{u_m}$, 
$$0 \le u - u_n \le A \left (\frac{B}{n}\right )^n \frac{1}{1 - \frac{B}{n}}.
$$
(Note that (*) above also gives another proof of the existence of the limit, by showing the sequence $(u_n)_{n = 1,2,3,...}$ to be Cauchy.)

The latter error bound is rather weak, but it suffices to prove, for example, that 
$$u = 1.8644589581634881323520037152739437841564220698266...$$
by computing $u_n$ for sufficiently large $n$. For example, $u_n$ has at least $n$ correct digits for any $n \ge 20$, because in that case 
$A \left (\frac{B}{n}\right )^n \frac{1}{1 - \frac{B}{n}} \le 0.5 \ 10^{-n}.
$ 

(Aside)
Here's the Sage program I used to compute the $u_n$:
def u(n):
  r = 0
  for k in [1..n][::-1]: r = sqrt(k*(k+1)/2 + r)
  return r

for n in [10..100]: print n, u(n).n(digits=100)

</s>