Raabe's test

Remarks

This is Kummer's test with $c_n = n$. Then $\rho_n = n a_n / a_{n+1} - (n + 1) = R_n - 1$, and $\sum 1/n$ diverges. For $R = 1$ the limit form gives no answer, and Bertrand's test can then help. Example: let $a_1 = 1/2$ and $a_{n+1} = a_n (2n+1)/(2n+2)$, so $a_n = \frac{1 \cdot 3 \cdots (2n-1)}{2 \cdot 4 \cdots (2n)}$. The ratio test fails, since $a_{n+1}/a_n \to 1$. Here $R_n = n/(2n+1) \le 1$, so $\sum a_n$ diverges. For $a_n = 1/n^p$ one gets $R_n \to p$, which recovers the p-series test for $p \ne 1$.

Going deeper

For a student

The ratio test compares a series with a geometric series. Raabe's test compares it with the $p$-series $\sum 1/n^p$ instead, through Kummer's test with weights $c_n = n$. It settles many series whose terms are quotients of products, such as binomial and hypergeometric coefficients. Gauss's test is a convenient form when $a_n / a_{n+1}$ has an expansion in powers of $1/n$.