Remark 2

The criteria of the second kind can be applied only to series whose general term behaves in a very regular way as $n$ tends to $+\infty$; in other words, their scope is much more restricted than those of the logarithmic criteria, and it would be a blunder to try to use them beyond the special cases to which they are particularly suited. For example, for the series $(u_n)$ defined by $u_{2m} = 2^{-m}$, $u_{2m+1} = 3^{-m}$ one has $u_{2m+1}/u_{2m} = \left(\frac{2}{3}\right)^m$, $u_{2m+2}/u_{2m+1} = \frac{1}{2} \left(\frac{3}{2}\right)^m$; the first of these ratios tends to 0 and the second to $+\infty$ as $m$ increases indefinitely, so no criterion of the second kind is applicable; nevertheless, since $u_n \preccurlyeq 2^{-n/2}$, it is immediate that the series converges.

Even when $u_{n+1}/u_n$ has a simple expression, a direct evaluation of a principal part for $u_n$ often leads to a result as quickly as the criteria of the second kind. For example, for the hypergeometric series, Stirling’s formula shows immediately that $u_n \sim a n^{\alpha + \beta - \gamma - 1}$, where $a$ is a constant $\neq 0$, and the logarithmic criterion of order 0 is then applicable.