TITLE: Product of Infinite series cubes
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: Hint. One may use a telescoping product, by writing as $n \to \infty$,
$$
\begin{align}
\prod_{k=7}^n\frac{k^3-1}{k^3+1}&=\prod_{k=7}^n\frac{k-1}{k+1} \cdot \prod_{k=7}^n\frac{k(k+1)+1}{k(k-1)+1}
\\&=\prod_{k=7}^n\frac{k-1}{k} \cdot \prod_{k=7}^n\frac{k}{k+1} \cdot \prod_{k=7}^n\frac{k(k+1)+1}{k(k-1)+1}
\\&=\frac6{n}\cdot \frac7{n+1} \cdot \frac{n(n+1)+1}{43}
\\&\to ?
\end{align}
$$