Telescoping product

Calculus Level 2

\[ A(x)=x+\cfrac { 1 }{ x+\cfrac { 1 }{ x+\cfrac { 1 }{ \ddots } } } \] With \(A(x)\) defined as above, where the continued fraction goes on indefinitely, find the value of the infinite product \[ \dfrac { 1 }{ A(1) } \times \dfrac { 1+\frac { 1 }{ 1 } }{ A(1) } \times \frac { 1+\frac { 1 }{ 1+\frac { 1 }{ 1 } } }{ A(1) } \times \cdots .\] If your answer can be expressed as \(\dfrac { a+\sqrt { b } }{ c\sqrt { b } }, \) where \(a, b, c\) are positive integers and \(b\) is square-free, give your answer as \(100a+10b+c\).


Bonus: Can you give a closed formula for \(\dfrac { y }{ A(y) } \times \dfrac { y+\frac { 1 }{ y } }{ A(y) } \times \dfrac { y+\frac { 1 }{ y+\frac { 1 }{ y } } }{ A(y) } \times \cdots \) when it converges?

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