Let $f(x)$ be a continuous function that converges to $0$ and can be differentiated for an infinite amount of times. Then we have $$f(x)=\sum_{k=1}^\infty\left(\frac{(-1)^kB_{2k}}{(2k)!}f^{(k+1)}(x)-f'(x+k)\right)-\frac{f'(x)}2$$ Where $B_k$ are the Bernoulli coefficients
The proof of this formula is straightforward. Take the derivative on both sides of the Euler-Maclaurin Summation formula and $$\sum_{k=1}^x f(k)=\sum_{k=1}^\infty(f(k)-f(k+x))$$ (which only works when $f(k)\sim0$), set the results equal to each other, and solve for $f(x)$. I am wondering if this formula has been discovered already, which I think it is because of its simple proof. If so, what is it called?
Edit: Since $f(x)$ is decreasing, this formula is pretty useless so it is possible that it isn't in the literature. I don't know for sure though.