Let $z \in \mathbb{C}$ with $|z|=1$ and $z\ne 1$. Now consider the sum $$S(N,p)=\sum_{k=0}^N k^p z^k,$$ for some positive integers $N,p$.

An immediate upper bound on $|S(N,p)|$ is $$|S(N,p)|\le C_1(p)N^{p+1},$$ for some constant $C_1$ depending only on $p$. I'm looking for a reference showing that accounting for the cancellation we have $$|S(N,p)|\le C_2(|1-z|,p)N^{p},$$ for some explicit constant $C_2$ depending only on $p,|1-z|,$ and monotonic decreasing in $|1-z|$.

It is possible to prove such a bound using $\sum_{k=0}^n k^p z^k = \left(z \frac{d}{dz}\right)^p \frac{1-z^{N+1}}{1-z}$, but since it's surly known I'm hoping for a reference.

Answers are much appreciated.


We may simply write $$S(n,p)(1-z)=-N^pz^{N+1}+\sum_{k=1}^{N} z^k(k^{p}-(k-1)^p),$$ thus by triangle inequality $$ \left|S(n,p)\right|\cdot |1-z|\leqslant 2N^p. $$


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