Closed form of infinite product $ \prod\limits_{k=0}^\infty \left(1+\frac{1}{2^{2^k}}\right)$

Question

What will be the value of the following Infinite Product :

$$\displaystyle \prod_{k=0}^\infty \left(1+\dfrac{1}{2^{2^k}}\right)$$

It would be nice if anyone could spare the time and boil down to the absolute basics and tell how they reached the solution.

Answer 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".

Answer 2

Alternative hint: The is like an Euler product for the sum $\sum_{n\geq 0} 2^{-n}$ with a binary expansion of $n$.

Answer 3

HINT: Multiply $1-\frac{1}{2^1}$ at the front.