To calculate the infinite product $$\prod_{n=1}^{\infty}\frac{2}{1+\pi^{\frac{1}{2^n}}},$$ I found that $\sum_{n=1}^{\infty}\ln\left(\frac{2}{1+\pi^{\frac{1}{2^n}}}\right)$ converges, so as $\prod_{n=1}^\infty\frac{2}{1+\pi^{\frac{1}{2^n}}}.$
The task though is asking to calculate the product and not just verify the convergence. Is there a mathematical way to calculate this product, or calculate this limit : $$\lim_{n\rightarrow\infty}\frac{2^n}{\left(1+\pi^{\frac{1}{2}}\right)\left(1+\pi^{\frac{1}{2^2}}\right)\cdots\left(1+\pi^{\frac{1}{2^n}}\right)}$$
Thank you for your time
