Suppose that $f[x]$ is an infinite sequence taking values in $[0, 1]$ (i.e. it is bounded and non-negative). We want to look at the sequence of partial sums:
$F[x] = \sum_{n=1}^x f[n]$
Let's also define $F_p[x]$ as the sequence of partial sums of the $p$'th powers of the sequence:
$F_p[x] = \sum_{n=1}^x f[n]^p$
Suppose we know the growth rate of $F[x]$ in terms of two functions:
$F[x] = g(x) + O(h(x))$
Can we thus extrapolate to get the growth rate of $F_p[x]$?