Positivity of a series of functions involving double poles constrained by certain inequalities This is a calculation I need for my statistics project
Big edit: simplify the function $f(x)$ a lot.
Define for $f(x)$, $x\geq 0$,
$$
f(x):=\sum_{k=1}^\infty \frac{a_k^2\lambda_k^2x}{(1+x\lambda_k)^2} - \sum_{k=1}^\infty \frac{b_k^2\beta_k}{(1+x\beta_k)^3}
$$
where $a_k\in \mathbb R$, $b_k\in\mathbb R$, $\lambda_k>0$, $\beta_k> 0$, and
$$
\sum_{k=1}^\infty b_k^2\leq \sum_{k=1}^\infty a_k^2<\infty\,\text{ and }\sum_{k=1}^\infty a_k^2 \lambda_k\leq \sum_{k=1}^\infty b_k^2 \beta_k<\infty.
$$
Additional assumption: we may think each $\beta_k$ is very large. You may take it as large as you want.
I am trying to prove that $f(x)$ has following graph. That is, prove that there exists $x_0>0$ such that $f(x_0)=0$, and for all $x<x_0$, $f(x)<0$, and for all $x>x_0$, $f(x)>0$.


This question has already been solved in this link We only need to take $\beta:=\min(\beta_i)$ and the function $(1+\beta x)^3f(x)$ is increasing.
 A: @Tankonetoone: On an experimental basis, I am convinced that the assertion is NOT TRUE, under hypothesis $\sum \lambda_k \leq \sum \beta_k < \infty $. BUT, it LOOKS TRUE (for all the cases I have tried...), if one assumes the inequality $\sum \beta_k \leq \sum \lambda_k < \infty $.
Indeed, I have taken at first $\lambda_k=\dfrac{1}{k^3}$ and $\beta_k=\dfrac{1}{k^2}$ in order that the initial summation inequalities are fulfilled.
I have made (see Matlab program below) a summation  with $1000$ (and even much more) terms, for various values of $\alpha$ in the range $(0,10000)$: the curve of function $f$, though increasing, remains desperately below the abscissas' axis.
I have then tried different other cases, e.g., $\lambda_k = \dfrac{1}{k^{2}}$ and $\beta_k = \dfrac{1}{k^{3/2}}$ all with the same negative result.
Reversing the roles of the $\lambda_k$s and $\beta_k$s, i.e., assuming $\sum \lambda_k \geq \sum \beta_k$, I obtained curves like the one displayed by the author in the different cases I have tried.
Could you confirm?

MATLAB PROGRAM:

clear all;close all;n=1000;
     L1=10;L2=500;step=0.5;
     range=L1:step:L2;
     Inv=1./(1:n);
     B=Inv.^2;L=Inv.^3;% not convenient
     %B=Inv.^3;L=Inv.^2;% convenient
     f=@(a)(sum((a*(L.^2)./((1+aL).^3)-B./((1+aB).^3))));
     S=[];
      for a=range;
         S=[S,f(a)];
      end; 
      plot(range,S)

