# Non-decreasing set of sequences with floor function

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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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 $10^7$
Thanks for the tighter bounds. It's starting to look like $x = 2$, but that's just a conjecture. – DPenner1 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