Converging infinite series

[Zuy]

Question

Let [MATH](u_n)[/MATH] be a sequence of real numbers that satisfy the following properties:

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

What can be said about the convergence of [MATH]\sum u_n[/MATH]?

Thoughts

There are 3 possible answers: every such [MATH]\sum u_n[/MATH] converges, every such [MATH]\sum u_n [/MATH] diverges, or both cases are possible.
The first one can be excluded, as [MATH](u_n)[/MATH] with [MATH]\forall n \in \mathbb{N}_{>1}: u_n=\frac{1}{n-1}[/MATH] 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?

 

[tkhunny]

"I was told" is always a problem. By whom? Why did you believe this individual?

Restate the Ratio Test.

 

[Zuy]

"I was told" is always a problem. By whom? Why did you believe this individual?

Restate the Ratio Test.

The professor thinks both cases are possible. Simultaneously, he wasn't able to come up with a converging example.

The ratio test gives [MATH]1^-[/MATH] and is therefore inconclusive.