I need to find example of a convergent and a divergent series that verify $\frac{U_{n+1}}{U_n} = 1 - \frac{1}{n} + o(\frac{1}{n})$ where $$U_n$$ is the general term of the series
For the divergent series I found $U_n = \frac{1}{n-1}$ But I don't have any idea for the convergent series I tried to simplify the expression but no result Any help would be appreciated
Thanks in advance
The idea is to find something "smaller" (decreasing faster) than $\frac{1}{n}$ so that the series converges, but not as small so as to mess up the approximately polynomial ratio. Try $U_n=\frac{1}{(n-1)\log^2(n-1)}$.The series converges by the integral test, and: $$ \frac{U_{n+1}}{U_n}=\frac{(n-1)\log^2(n-1)}{n\log^2 n}=(1-\frac{1}{n})(\frac{\log(n-1)}{\log n})^2 $$ with $$\frac{\log(n-1)}{\log n}=1+\frac{-\frac{1}{n}+o(\frac{1}{n})}{\log n}=1+o(\frac{1}{n})$$ by Taylor expansion.