Convergence depending on the parameter

Question

Let $c\ge 0$ be a real number. Then we define $$a_1=1, \quad a_{n+1}=\frac{cn+1}{n+3} a_n$$ Investigate convergence of $\displaystyle \sum_{n=1}^{\infty} a_n$ depending on the parameter $c$.

Here I am also completely lost how to investigate convergence if I do not know the general term ? How to approach this?

Answer 1

Hint: Take cases depending on whether $$\frac{cn+1}{n+3}<1,\quad \frac{cn+1}{n+3}>1 \quad \text{ or } \frac{cn+1}{n+3}=1$$

Answer 2

Use the ratio test and look at $a_n/a_{n+1}=(cn+1)/(n+3)$. The limit of this ratio as $n\to\infty$ is $c$. If $c<1$, then the series converges; if $c>1$, then the series diverges. What happens if $c=1$?