# Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for

$$\lim_{n\to-\infty}\sum_{i=n}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)}$$

The series has the form:

$$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + \frac{x^{\frac{1}{2}}(x^{\frac{1}{2}}-1)}{x^{2}+1} + \frac{x(x-1)}{x^4+1} +\frac{x^2(x^2-1)}{x^8+1} ...$$

Among other things the closed form satisfies

$$g(x) = g\left(\sqrt{x}\right)$$

where we select the principal square root.

• Does this even converge? – Yuriy S May 23 at 12:38