O-notation property - sum of the first n powers growth I read here that in the tenth property: < The sum of the first $nr^{th}$ powers grows as the $(r+1)^{th}$ power This is not very intuitive to me, why is that? Since the base is changing (not the exponent) in the summatory I can't solve it with a geometric series

For intuition, think about converting your sum to an integral. $\sum_{i=0}^n i^r \approx \int_0^n i^r di=\frac 1{r+1}n^{r+1}$

If you try it for small values, for $r=0$ you get the sum of $n \ 1$'s, which is $n$ (growing like $r+1=1$). For $r=1$ you get the triangular numbers $\frac {n(n+1)}2$, (growing like $r+1=2$).