This is just a curiousity question.
HISTORY:-
$Motivation:$
I first started wondering about this question about 3 months ago when I first got interested in continued fractions but I really didn't tried to get its value but when I saw the thumbnail of this video https://youtu.be/CaasbfdJdJg, I quickly clicked and after watching the whole video I realized that the thumbnail was a click bait and the host didn't even talked about the thing in the thumbnail.
$Introduction:$
After the biggest anime betrayal of my life from one of my favourite math channel, I tried to get the value of $1+\frac{1}{2+\frac{1}{3+\dots}}$ by myself even though I didn't really have enough mathematical knowledge and as you would expect I really got nowhere and after some sad math noises I forgot about the expression for nearly 2 months but suddenly yesterday I remembered the thing again and here I am asking this question.
THE NUMBER:-
Let $\textstyle\displaystyle{Q=1+\frac{1}{2+\frac{1}{3+\ddots}}}$
Obviously whatever $Q$ is, it is an irrational number because the continued fraction is infinite.
After playing with wolfram alpha it is clear that $1.4<Q<1.5$ is true.
It also seems compelling that $Q$ does converge to a specific real value because
If we define $\textstyle\displaystyle{Q_n=1+\frac{1}{2+\frac{1}{3+\dots\frac{1}{n}}}}$
Then, $\textstyle\displaystyle{Q=\lim_{n\rightarrow\infty}(Q_n)}$
$\textstyle\displaystyle{|Q_1-Q_2|=\frac{1}{2}}$
$\textstyle\displaystyle{|Q_2-Q_3|=\frac{1}{14}}$
$\textstyle\displaystyle{|Q_3-Q_4|=\frac{1}{210}}$
So it seems, $\textstyle\displaystyle{\lim_{n\rightarrow\infty}(Q_n-Q_{n-1})=0}$
Finally the question:-
I am not much of a mathematician, so please for the mathematician folks out there I would like to know the closed form of $Q$ if there exists one and also the derivation of it or some bounding. (The previous bounding was not derived but rather observed from wolfram alpha)