Kummer's test

Remarks

For part 1, multiply $\rho_n \ge \delta$ by $a_{n+1}$ to get $c_n a_n - c_{n+1} a_{n+1} \ge \delta a_{n+1}$ for $n \ge N$. Summing from $N$ to $m$ telescopes, so $\delta \sum_{n=N+1}^{m+1} a_n \le c_N a_N$, and the partial sums are bounded. For part 2, the sequence $c_n a_n$ increases from $n = N$ on, so $a_n \ge c_N a_N / c_n$, and the comparison test with $\sum 1/c_n$ gives divergence. The choice $c_n = 1$ gives the ratio test, the choice $c_n = n$ gives Raabe's test, and the choice $c_n = n \ln n$ gives Bertrand's test.

Going deeper

For a student

When the ratio test gives the limit $1$, one uses Kummer's test with weights that grow, such as $n$ or $n \ln n$. Each such choice gives a finer test: Raabe's test catches the $p$-series, and Bertrand's test catches series like $\sum 1/(n (\ln n)^p)$. The proof only uses telescoping sums and the comparison test.