Let the function $f_{n}:\mathbb{R}\rightarrow \mathbb{R_{\ge 0}}$ be defined as $$ f_{n}(x)=\sum_{i,j=1}^{n}\frac{(-1)^{i+j}\cos(\ln \frac{i}{j})}{(ij)^{x}}\quad \forall n\in\Bbb N $$ There is given two different positive values: $x_{1},x_{2}$.
For given natural $N$ find possibly the best lower bound of sum $F(N):=f_{N}(x_{1})+f_{N}(x_{2})$
And also:
Find limit inferior of $F(n)$
How to in general solve that kind of problems ?(i mean to find limit inferior of sum of functions)
Regards.
EDIT: My idea was to checking every pair of consecutive elements of this sequence i.e to find $$\max\{F(N),F(N+1)\}\quad \forall n\in\Bbb N$$ Is this true that $\inf F(N)>0$ ?
What if i ask about positivity of limit of $F(n)$ ?