Is the infinity product $\prod_{k=1}^\infty\frac{1+a^k}{1+a^{k+x}}$ convergent?

Question

Let $a$ be a real number belong to $(0,1)$ and consider the following infinity product
$$\prod_{k=1}^\infty\frac{1+a^k}{1+a^{k+x}}$$ Is the above product convergent( for which x)? does it have a closed form?

Answer 1

In the comments someone said that the product converges for $x>0$ so, for $x<0$, put $y=x-[x]+1>0$. $$\prod_{k=1}^{n}\frac{1+a^{k}}{1+a^{k+x}}=$$ $$ \prod_{k=1}^{-[x]+1}\frac{1}{1+a^{k+x}}\cdot\prod_{k=1}^{n+[x]-1}\frac{1+a^{k}}{1+a^{y+k}}\cdot\prod_{k=n+[x]}^{n}(1+a^{k}) $$ Since $a\in(0,1)$, the product converges.