[SOLVED] Convergence/divergence of an integral

Hi again !

This time, I'm really struggling with a problem...

**Text**

Let $\displaystyle C_0(\mathbb{R}^+)$ be the set of functions defined over $\displaystyle \mathbb{R}^+$, continuous, with real values.

$\displaystyle f \in C_0(\mathbb{R}^+)$. Let F be the antiderivative of f that annulates in 0.

Let E be the subset of functions f in $\displaystyle C_0(\mathbb{R}^+)$ such that $\displaystyle I(f)=\int_0^\infty \frac{F(t)}{(1+t)^2} \ dt$ converges.

**Previous questions...**

1/ Determine the positive functions f in E such that $\displaystyle I(f)=0$.

It's ok, f=0.

2/ Let f be a **positive **function of $\displaystyle C_0(\mathbb{R}^+)$

Show : $\displaystyle \int_0^\infty \frac{f(t)}{1+t} \ dt \text{ converges } \Longleftrightarrow f \in E$

It's ok for this.

**Problem :**

Find an example for f (necessarily not of constant sign) in E and such that $\displaystyle \int_0^\infty \frac{f(t)}{1+t} \ dt$ diverges.

Thanks for your help, we are really struggling with that :(

(and I have no assignment, I'm asking several questions these days because I have an exam tomorrow :))