Bertrand's test

Remarks

This is Kummer's test with $c_n = n \ln n$. A short computation gives $\rho_n = B_n - (n+1) \ln(1 + 1/n)$. The inequality $\ln(1 + x) \ge x/(1 + x)$ gives $(n+1)\ln(1 + 1/n) \ge 1$, and this quantity tends to $1$. So $B_n \ge r > 1$ gives $\rho_n \ge (r - 1)/2$ for large $n$, and $B_n \le 1$ gives $\rho_n \le 0$; moreover $\sum 1/(n \ln n)$ diverges. Example: if $a_n / a_{n+1} = 1 + 1/n + O(1/n^s)$ with $s > 1$, then $B_n = \ln n \cdot O(n^{1-s}) \to 0$, so the series diverges. This is the case that Raabe's test leaves open in Gauss's test. For $a_n = 1/(n (\ln n)^p)$ one finds $B_n \to p$, so the test decides these series for $p \ne 1$.

Going deeper

For a student

The ratio test, Raabe's test and Bertrand's test form a scale of finer and finer tests, all special cases of Kummer's test. Each one looks at the next term in the expansion of $a_n / a_{n+1}$. One use of Bertrand's test is the proof of Gauss's test in the boundary case $h = 1$.