Is it true?: $\prod_{n=1}^\infty \frac{1}{1+2^{1/2^n}}=\ln 2$

Question

I found another amazing memo from my old note!
It says $$\prod_{n=1}^\infty \frac{2}{1+2^{1/2^n}}=\ln 2$$ Since the partial product for $>1000$ terms is $0.69314718056 \approx \ln 2$, it seems to be true.
Unfortunately, I didn't wrote why it is true.
I combed all over my bookshelf but there was no additional clues.
Would you help me?

Answer 1

Show by induction that: $$\prod_{k=0}^{N-1}\left(1+x^{2^k}\right)=\frac{x^{2^{N}}-1}{x-1}$$ This is just a result of repeated factoring of difference of two squares.

Letting $x=2^{1/2^N},$ and $j=N-k,$ $$\prod_{j=1}^N\left(1+2^{1/2^j}\right)=\frac{1}{2^{1/2^N}-1}$$

So $$\prod_{j=1}^N\frac{2}{1+2^{1/2^j}}=2^N\left(2^{1/2^N}-1\right)$$

Letting $u=1/2^N$ compute $$\lim_{u\to 0}\frac{2^u-1}{u}.$$

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In general, for $a>0, a\neq 1,$

$$\prod_{j=1}^\infty\frac{2}{1+a^{1/2^j}}=\frac{\ln a}{a-1}$$