I find the following series but don't know if it's true. For $p=0,1,2,3,\cdots$, $$ \lim_{n\to\infty} \frac{1}{n}\sum_{i=1}^n\left\{\left(\sum_{j=i}^n\frac{1}{j}\right)^p\right\} = p! $$ It's easy to verify for $p=0$ or $1$. For $p \ge 2$, I tried to evaluate the series using this piece of R code here, and the result looks promising.
par(mfrow = c(3, 3))
n <- 1:1000 * 100
f <- function(n, p){ mean(cumsum(1/(n:1))^p) }
f <- Vectorize(f)
for (p in 1:9){
plot(n, f(n, p))
abline(h = factorial(p))
}
Does this series look like anything familiar to you? Can you help me prove this, or give me some suggestions?
Your conjecture is not difficult to prove, since
$$ \sum_{j=i}^{n}\frac{1}{j}=H_n-H_{i-1}=-\log\left(\frac{i-\frac{1}{2}}{n+\frac{1}{2}}\right)+O\left(\frac{1}{i^2}+\frac{1}{n^2}\right) $$ and by Riemann sums $$ \frac{1}{n}\sum_{i=1}^{n}\left[-\log\left(\frac{i-\frac{1}{2}}{n+\frac{1}{2}}\right)\right]^p \to \int_{0}^{1}\left(-\log x\right)^p\,dx=p!.$$