Product of Infinite series cubes

Question

If $$X=\left({342\over 344}\right)\left({511\over 513}\right)\left({728\over 730}\right)\dots$$ Up to infinite terms.. The value of $x$ approaches? What's the approach to the above problem? They can be expressed as cubes-1/cubes +1.how to simplify further?

Answer 1

Hint. One may use a telescoping product, by writing as $n \to \infty$, $$ \begin{align} \prod_{k=7}^n\frac{k^3-1}{k^3+1}&=\prod_{k=7}^n\frac{k-1}{k+1} \cdot \prod_{k=7}^n\frac{k(k+1)+1}{k(k-1)+1} \\&=\prod_{k=7}^n\frac{k-1}{k} \cdot \prod_{k=7}^n\frac{k}{k+1} \cdot \prod_{k=7}^n\frac{k(k+1)+1}{k(k-1)+1} \\&=\frac6{n}\cdot \frac7{n+1} \cdot \frac{n(n+1)+1}{43} \\&\to ? \end{align} $$