Converging infinite series

Zuy

New member
Joined
May 16, 2019
Messages
2
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
Joined
Apr 12, 2005
Messages
9,931
"I was told" is always a problem. By whom? Why did you believe this individual?

Restate the Ratio Test.
 

Zuy

New member
Joined
May 16, 2019
Messages
2
"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.
 
Top