[Math] Compute the infinite product $\prod\limits_{n=2}^\infty \left(1+\frac{1}{2^n-2}\right)$

Question

I am trying to compute the infinite product

$$
\prod\limits_{n=2}^\infty \left(1+\frac{1}{2^n-2}\right) .
$$

Wolfram Alpha says the result is $2$, but I can't seem to figure out why.

Answer 1

HINT:

$$1+\frac1{2^n-2}=\frac{2^n-1}{2(2^{n-1}-1)}$$

Clearly, the denominator of each term gets cancelled by the numerator of the previous except for the last term

$$\implies\prod_{n=2}^N\left(1+\frac1{2^n-2}\right)=\frac{2^N-1}{2^{N-1}}$$