So I am preparing for my math test and as a practice try to crack this problem
Does the following series $ \sum_{n=1}^{\infty} a_n $ converge or diverge? $$ a_1 = 3, \quad a_{n+1} = \left( \frac{n}{n+1} \right) a_n $$
We were taught a bunch of different methods such as limit comparison, ratio test, root test and etc. However, this series is recursive so I am not quite sure how to approach it. One thing I thought about is replacing $a_n$ and $a_{n+1}$ with $L$ and then calculate $L$. However, I have $n$ variable here which kind of screws everything up. Could you please guide me through the solution or at least tell me how I can approach it.
Hint: $\;(n+1)\cdot a_{n+1}=n \cdot a_n = (n-1) \cdot a_{n-1}=\ldots=1 \cdot a_1$, then think at $\,\sum_{n \ge 1} \dfrac{1}{n}\,$.