Converging infinite series

Zuy

New member
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
• $$\displaystyle \frac{u_{n+1}}{u_n}=1-\frac{1}{n}+O\left(\frac{1}{n^2}\right)$$
What can be said about the convergence of $$\displaystyle \sum u_n$$?

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?

tkhunny

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

Restate the Ratio Test.

Zuy

New member
"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 $$\displaystyle 1^-$$ and is therefore inconclusive.