How can this expression be calculated? $\frac{1+{\displaystyle \frac{3\cdots}{4\cdots}}}{2+{\displaystyle \frac{5\cdots}{6\cdots}}}$

I can't see any obvious way this could be calculated. It seems to converge to a value of approximately 0.6278...

$\frac{1+{\displaystyle \frac{3}{4}}}{2+{\displaystyle \frac{5}{6}}}}\approx 0.6176$

$\frac{1+{\displaystyle \frac{3+{\displaystyle \frac{7}{8}}}{4+{\displaystyle \frac{9}{10}}}}}{2+{\displaystyle \frac{5+{\displaystyle \frac{11}{12}}}{6+{\displaystyle \frac{13}{14}}}}}}\approx 0.6175$

Going all the way up to 62 gives a result of 0.627841944566, so it seems to converge.

Is it possible to find a value for this? Will it have a closed form solution?

I can't see any obvious way this could be calculated. It seems to converge to a value of approximately 0.6278...

$\frac{1+{\displaystyle \frac{3}{4}}}{2+{\displaystyle \frac{5}{6}}}}\approx 0.6176$

$\frac{1+{\displaystyle \frac{3+{\displaystyle \frac{7}{8}}}{4+{\displaystyle \frac{9}{10}}}}}{2+{\displaystyle \frac{5+{\displaystyle \frac{11}{12}}}{6+{\displaystyle \frac{13}{14}}}}}}\approx 0.6175$

Going all the way up to 62 gives a result of 0.627841944566, so it seems to converge.

Is it possible to find a value for this? Will it have a closed form solution?