Problem 182 (difficulty: 4/10)

Give examples of sequences \(\displaystyle a_n\), with the property \(\displaystyle \displaystyle\frac{a_{n+1}}{a_n}\to1\), such that

 (1)  \(\displaystyle a_n\) is convergent;      (2)  \(\displaystyle a_n\to\infty\);  (3)  \(\displaystyle a_n\to-\infty\);      (4)  \(\displaystyle a_n\) is oscillating.

Give me another random problem!