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_{5, 1.1} $ represents the 5th term in the sequence where $ c = 1.1 $. Since $ S_{5, 1.1} = 2 $ and $ S_{6, 1.1} = 1 $, this is an example of a decreasing sequence.
For what values of $c$, then, is the sequence non-decreasing?
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!
Edit: My original question asked just for the following two specific cases, which are still the most interesting cases for me, and would still be helpful if a general solution cannot be obtained:
- 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 \ne 1 $ for which it is non-decreasing?
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 $10^7$