<s>[INST] Limit of $\frac 12$, $\frac{1\cdot 4}{2 \cdot 3}$, $\frac{1\cdot 4\cdot 5}{2\cdot 3\cdot 6}$, ... The following is not a homework, just curiosity.
Consider the integers grouped by consecutive pairs : $(1,2)$, $(3,4)$, ...
What is the limit of the "switching fractions" where we alternatively use the largest number in a pair either upward or downward :
$$\frac 12, \frac{1\cdot 4}{2 \cdot 3}, \frac{1\cdot 4\cdot 5}{2\cdot 3\cdot 6}, \frac{1\cdot 4\cdot 5\cdot 8}{2\cdot 3\cdot 6\cdot 7},\ldots?$$
A proof as elementary as possible would be nice, if not it could use standard results on prime distribution.
Also, was it considered before? Any reference welcomed.
Edit
Numerically we have:
$0.5, 0.6666... , 0.5555... , 0.6349206..,0.5714286..,0.6233766...$ More terms would certainly help.
 [/INST] A more simpler, yet heavier way to look at this is,
A more common series is given by $\dfrac{1}{2}\cdot\dfrac{3}{4}\cdot\dfrac{5}{6}\cdot\ldots=\prod\limits^{n}_{k=1}{\dfrac{2k-1}{2k}}=\dfrac{\left(n-\frac{1}{2}\right)!}{n!\cdot\left(\frac{-1}{2}\right)!}$.
Your problem is slight altered with every alternate term inversed, $\dfrac{1}{2}\cdot\left(\dfrac{3}{4}\right)^{-1}\cdot\dfrac{5}{6}\cdot\left(\dfrac{7}{8}\right)^{-1}\cdot\ldots$
In fact we factor with respect to the inverse power, $$\left(\frac{1}{2}\cdot \frac{5}{6}\cdot\ldots\right)\cdot\left(\frac{3}{4}\cdot\frac{7}{8}\cdot\ldots\right)^{-1}$$
$$=\left(\prod^{n}_{k=1}{\frac{4k-3}{4k-2}}\right)\cdot\left(\prod^{n}_{k=1}{\frac{4k-1}{4k}}\right)^{-1}$$
$$=\left(\frac{\left(n-\frac{1}{4}\right)!}{n!\cdot \left(\frac{-1}{4}\right)!}\right)^{-1}\cdot\frac{\left(n-\frac{3}{4}\right)!\cdot\left(\frac{-1}{2}\right)!}{\left(n-\frac{1}{2}\right)!\cdot\left(\frac{-3}{4}\right)!}$$
Funnily enough, the limit does converge for the above expression ($n\to\infty$); $$\frac{\left(\frac{-1}{4}\right)!\cdot \left(\frac{-1}{2}\right)!}{\left(\frac{-3}{4}\right)!}\approx 0.599195 \text{ as mentioned in answers above}$$
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