= Polylogarithm series remainder bound
{title2=$|R_N|\leq |z|^{N+1}/[(N+1)^a(1-|z|)]$}
For $|z|<1$, $a\geq0$ and integer $N\geq0$, truncating the defining <polylogarithm> <power series> after $N$ terms has remainder
$$
\left|\sum_{k>N}\frac{z^k}{k^a}\right|
\leq\sum_{k>N}\frac{|z|^k}{(N+1)^a}
=\frac{|z|^{N+1}}{(N+1)^a(1-|z|)}.
$$
The inequality uses monotonicity of $k^a$ and then the <geometric series>. The bound applies to negative real $z$ as well as positive real $z$, and gives a rigorous absolute truncation tolerance independently of cancellation. It does not bound floating-point rounding error.
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