For any positive integers $n,d$, let

$$ A_d(n)=\frac{\sum_{k=1}^n k^{2d}}{n(n+1)(2n+1)} $$

It is easy to see (and well-known) that for fixed $d$, $A_d(.)$ is a polynomial of degree $2d-2$. Then we can speak of $A_d(x)$ for any $x\in{\mathbb R}$, not just the positive integers.

Is it known for which real numbers $x$ the sequence $(A_d(x))_{d\geq 1}$ is bounded ?

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