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I recently solved a practical sequence problem, but got curious and tried to generalize it. Let

$$ S_{n, c} = \lfloor \left( n+1 \right)^c \rfloor - \lfloor n^c \rfloor $$

be a set of sequences where $ n \in \mathbb{N}, c \in \mathbb{R}^+ $, that is the positive reals. I'm not sure if my notation is clear, but as an example, $ S_{4, 1.5} $ represents the 4th term in the sequence where $ c = 1.5 $. I have two related questions on this set of sequences:

  • What is the smallest $ x \in \mathbb{R}^+ $ such that for all sequences $ S_{n, c} $ where $c \ge x, S_{n, c} $ is non-decreasing (if it exists)?
  • What is the smallest $ c $ for which it is non-decreasing? (Edit: I'm looking an interesting result excluding the possibility that $ c = 1 $)

An example of a sequence that has decreasing terms is one with $ c = 1.1 $. $ S_{5,1.1} = 2 $, while $ S_{6,1.1} = 1 $.

I've only ever had one class on sequences and series, and its been two years, so I'm quite rusty! Using the binomial theorem, I was able to prove that it's non-decreasing when $ c $ is an integer, but I've so far not been able to extend that. Any help is greatly appreciated!

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1 Answer 1

For the first bullet, $1.96 \lt x \le 2$, as $S_{7958491,1.96}=8261400,S_{7958492,1.96}=8261399$ and for $c \gt 2$ the difference without the floor signs is $\gt 2$. My program gets slow checking more than $10^7$

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Thanks for the tighter bounds. It's starting to look like $ x = 2 $, but that's just a conjecture. –  DPenner Mar 27 '13 at 0:35
    
@DPenner: For what it is worth, nothing found at $n=1.965$ under $50,000,000$ –  Ross Millikan Mar 27 '13 at 0:46

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