__Question__

Let \(\displaystyle (u_n)\) be a sequence of real numbers that satisfy the following properties:

- \(\displaystyle \forall n \in \mathbb{N}: u_n >0\), and
- \(\displaystyle \frac{u_{n+1}}{u_n}=1-\frac{1}{n}+O\left(\frac{1}{n^2}\right)\)

__Thoughts__

There are 3 possible answers: every such \(\displaystyle \sum u_n\) converges, every such \(\displaystyle \sum u_n \) diverges, or both cases are possible.

The first one can be excluded, as \(\displaystyle (u_n)\) with \(\displaystyle \forall n \in \mathbb{N}_{>1}: u_n=\frac{1}{n-1}\) is a diverging counter-example.

I was told that both cases were possible, but I've not been able to come up with a converging example.

Any ideas?