Say I have an infinite sequence $X=(x_i)$, $i=1,2,3,\ldots$ such that it's in $\ell^2$ space, i.e. $\sum_{i=1}^\infty|x_i|^2<\infty$. Now, this function that takes this infinite sequence to a real number:
$$\tag{1}F(X)=\sum_{i=1}^\infty\ln \left(1+\frac{x_i}{a_i}\right)$$
(let's assume that $x_i\geq0$ for all $i$ and (1) converges).
What can I say about the Frechet derivative at $X$? Does the fact that $\frac{\partial F}{\partial x_i}=\frac{1}{a_i+x_i}$ exist for every $i$ tell me anything? How do I compute it here?