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Is there another way to determine monotonicity besides $U_{n+1} - U_n$?

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U_{n+1} / U_n. More generally, you can compose U_n with any increasing function (this preserves monotonicity), then test the monotonicity of the new sequence. – Qiaochu Yuan Oct 30 '10 at 19:01
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(I'm in a silly mood, sorry) I find most sequences very dull, so for me the test would be $P(U_n$ is monotonous $) = 1$ almost always. – Andrew Stacey Oct 30 '10 at 19:03
Andrew is funny, but I'd lower the probability to $1-\epsilon$ where $\epsilon$ is exceedingly tiny. In any event, it's "monotonicity". – J. M. Oct 30 '10 at 22:54
Thanks hehe :P Duly edited – Qosmo Oct 30 '10 at 23:56

1 Answer

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I'll write something so this question doesn't remain unanswered.

Sometimes it can be quite difficult to prove that a sequence is increasing. For example, Smetaniuk gave a proof that the number of Latin squares $L_n$ (Sloane's A002860) is increasing (actually, he proved $L_{n+1} \geq (n+1)!L_n$), which is one of the best results around regarding the mysterious $L_n$.

Smetaniuk, Bohdan A new construction of Latin squares. II. The number of Latin squares is strictly increasing. Ars Combin. 14 (1982), 131–145.

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