Approximation of the Bernoulli periodic function

Question

I remember seeing a paper that provided a summation approximation of the Bernoulli periodic function which converges when $p\ge 2$;

$$\dfrac{P_{p}(x)}{(p!)}$$

but I don’t quite remember it,

I know for $p = 2; P_2(x)/2!$ it is $$\sum_{k=1}^{\infty} \dfrac{\cos[(2\pi k x - 2 \pi /2)]}{(2 k \pi)^2}$$

But I can’t quite remember the general expression for all $p\ge 2$.

Answer 1

Found it apparently the general form; is $$-2 \sum_{k=1}^{\infty} \dfrac{\cos[2 k \pi x - p \pi/2]}{(2 k \pi)^p}$$