Gauss's test

Remarks

The quantity of Raabe's test is $n(a_n/a_{n+1} - 1) = h + \beta_n n^{1-s}$, which tends to $h$. So Raabe's test settles the cases $h > 1$ and $h < 1$. For $h = 1$, the quantity of Bertrand's test is $\beta_n n^{1-s} \ln n$, which tends to $0$, so the series diverges. Example: let $a, b, c > 0$ and let $a_n$ be the $n$-th term of the hypergeometric series $\sum_n \frac{(a)_n (b)_n}{(c)_n \, n!}$, where $(x)_n = x(x+1)\cdots(x+n-1)$. Then $a_n / a_{n+1} = \frac{(n + c)(n + 1)}{(n + a)(n + b)} = 1 + \frac{c + 1 - a - b}{n} + O(1/n^2)$. So the series converges exactly when $c > a + b$.

Going deeper

For a student

Gauss's test packages Raabe's test and Bertrand's test into one statement for ratios with an asymptotic expansion. A classical application is the convergence of the hypergeometric series at $x = 1$, as in the example above. The error term must be $O(1/n^s)$ with $s > 1$; an error like $1/(n \ln n)$ would hide the boundary case.