Finding big O notation of function with two parameters
Some of modifications of O described in Concrete Mathematics: Foundation for Computer Science says:

$ \qquad O(f(n)) + O(g(n)) = O(\mid f(n)\mid + \mid g(n) \mid) \qquad (9.22)\\
  \qquad O(f(n))O(g(n)) = O(f(n)g(n)) \qquad \qquad \qquad (9.26)
$

And using some basic knowledge of O notation and functions:

$
\qquad O(f(n) +c) = O(f(n)) \\
\qquad \forall_{n\in \mathbf N} \forall_{k&gt;0} n^k &gt; 0
$

Use those transformation you can came up with something like this:

$$\frac{n^{s+1}-1}{n-1}-1 = O\left(\frac{n^{s+1}-1}{n-1}-1\right) = 
O\left(\frac{n^{s+1}-1}{n-1}\right) = 
O\left(\frac{1}{n-1}\right)O\left(n^{s+1}-1\right) =
$$
$$
O\left(\frac{1}{n}\right)O\left(n^{s+1}\right) = O\left(\frac{n^{s+1}}{n}\right) = O(n^s)
$$

So your intuition was right.