<s>[INST] Compute the infinite product $\prod\limits_{n=2}^\infty \left(1+\frac{1}{2^n-2}\right)$ 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. 
 [/INST] Write 
$$1 + \frac{1}{2^n - 2} = \frac{2^n - 1}{2^n - 2} = \frac{1}{2} \frac{2^n - 1}{2^{n-1} - 1}$$
so by telescoping,
$$\prod_{k = 2}^N \left(1 + \frac{1}{2^n - 2}\right) = \frac{1}{2^{N-1}}(2^N - 1) = 2 - \frac{1}{2^{N-1}}.$$
Since $\frac{1}{2^{N-1}} \to 0$ as $N \to \infty$, we have
$$\prod_{n = 2}^\infty \left(1 + \frac{1}{2^n-2}\right) = 2.$$
</s>