Closed form of infinite product $ \prod\limits_{k=0}^\infty \left(1+\frac{1}{2^{2^k}}\right)$ [closed]
Solution 1:
hint: $1+\dfrac{1}{2^n} = \dfrac{1-\dfrac{1}{2^{2n}}}{1-\dfrac{1}{2^n}}, n = 2^k$, and realize "product telescoping".
Solution 2:
Alternative hint: The is like an Euler product for the sum $\sum_{n\geq 0} 2^{-n}$ with a binary expansion of $n$.
Solution 3:
HINT: Multiply $1-\frac{1}{2^1}$ at the front.