Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
3  
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
5  
(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
add comment

1 Answer

up vote 2 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.