I have the following question:

Does there exist a non-negative function $g$ on $(0,1)$ such that $$1\leq F(x):=\dfrac{\displaystyle\sum_{k=0}^{\infty}a_{k}\,(k+1)^{2}\,x^{k}}{\displaystyle\sum_{k=0}^{\infty}(k+1)\,x^{k}}\leq 2,\;\forall\;x\in (0,1),$$ and $$\displaystyle\lim_{x\rightarrow 1^{-}}F(x)\;\text{ does not exist?}$$ Here $$a_{k}:=\int_{0}^{1}g(x)\,x^{2k+1}dx.$$

Note that if $g\equiv2$ then the first condition satisfies, but the second condition doesn't.